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BASIC GEOMETRY KEY CONCEPTS

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BASIC GEOMETRY KEY CONCEPTS Powered By Docstoc
					BASIC GEOMETRY KEY CONCEPTS
You will need the following to do this course: Geometer’s Sketchpad (GSP), a compass,
a straight edge, a calculator, a protractor, colored pencils, tape and Algebra I course
notes.

I.       Unit 1: The Language and Symbols of Geometry

         A. Unit 1 Introduction

The learner will investigate the language and symbols of geometry by utilizing the terms
of points, lines, planes, segment, midpoints, rays, angles, angle pairs, and perpendicular
bisectors, as well as analyze two-dimensional and three-dimensional figures and relate to
life-related problems.

The learner will:

        find angle relationships such as vertical angles, linear pairs, complementary
         angles, and supplementary angles.
        identify relationships between and among points, lines, and planes, such as
         betweenness of points, midpoint, distance, collinear, coplanar.....
        find the intersection of lines, planes, and solids.
        connect geometric diagrams with algebraic representations.
        integrate constructions such as segments and angles, segment bisectors, angle
         bisectors.
        use relationships among one-and two-dimensional measures.
        represent geometric figures and properties using coordinates.
        connect the concepts of distance and midpoint to coordinate geometry.


         B. Lesson 1: Points, Lines, Planes, and Space

Read the definitions of line, point, plane, collinear points, non-collinear, coplanar points,
non-coplanar points, betweeness of points and segment.

Use the Sketchpad (GSP) software and complete the following exercise. Print out your
work. If you're using the demo version, please draw on paper what you see on your GSP
screen.
Procedure:
    1. Select Segment Tool and draw segment AB.
    2. Use Select Tool and the shift key to select both end points.
    3. Click on the Display menu and choose show labels.
    4. Select the segment and go to the Construct menu and choose point on object.
        Again, use Display to show labels.
    5. Select two of the points using the shift key.
   6. Choose the Measure menu and choose distance to measure each distance from A
       to B, B to C and A to C.
   7. Select the segments AC and BC using the shift key.
   8. Choose the Measure menu and calculate to find the sum.
   9. Drag point C along segment AB while holding down the shift key.
   10. Go to Tool to record what you observed.
   11. Go to the Edit menu and select all.
   12. Go to the File menu and print.

Complete this problem. Given point D is between points T and R. TD = 3x + 2, DR = 2x
+ 1, and TR = 38. Find TD and RD. Show work. (Hint: Draw a diagram first.)

       C. Lesson 2: Distance and Midpoint

Read the definitions for distance and midpoint.
Now, you will use those concepts and the distance and midpoint formulas to solve some
problems.
Distance formula: d             x2  x1    y2  y1 
                                          2                 2



                         x  x y  y2 
Midpoint formula: M   1 2 , 1        
                         2        2 
Examples: Given the coordinates of A and B, find the distance AB and the midpoint M of
segment AB.
   1. A: (7, 11) and B: (1, 3)
                                                 7  1 11  3 
      d  1  7    3  11             M 
                   2           2
                                                        ,      
                                                 2        2 
                                                 8 14 
      d   6    8                   M  , 
                 2        2

                                                2 2 
      d  36  64                          M   4, 7 
       d  100
       d  10
       So, AB = 10 and M = (4, 7)

   2. A: (1, 2) and B: (4, 6)
                                                                 1 4 2  6 
       d     4  1   6  2                            M 
                    2                 2
                                                                        ,    
                                                                 2        2 
                                                                5 8
       d     3   4                                    M  , 
                2           2

                                                                 2 2
       d  9  16                                           M   2.5, 4 
       d  25
       d 5
       So, AB = 5 and M = (4, 7)
Complete the following problems: given the coordinates of A and B, find AB, the
coordinates of M, and the midpoint of segment AB.
                   a) A: (4, 2)             b) A: (-9, -1)              c) A: (-2, 1)
                      B: (7, 0)                B: (-6, -2)                 B: (5, -3)
            AB = __________            AB = __________           AB = __________
                M: _________              M: _________               M: _________


       D. Lesson 4: Perpendicular

Read the definitions for perpendicular lines, perpendicular bisectors, and midpoints.
Now you will construct the perpendicular bisector of a line segment. To do this you will
need a compass and straight edge.
Procedure:
   1. Draw a line segment.


   2. Draw a circle that has one of the line segment endpoints as its center and whose
      radius is more than half the length of the line segment.
   3. Draw another circle with the other endpoint as its center and whose radius is more
      than half the length of the line segments.




   4. Draw a line through the two points where the circles intersect. This is the
      perpendicular bisector.




   (Where the perpendicular bisector and line segment meet is the midpoint.)

Then, complete the following constructions:

   1. Draw a segment with your straight edge. Its length doesn't matter. Label the
      segment AB.
   2. Construct a line perpendicular to segment AB that does NOT pass through the
      midpoint. Label the intersection point X.
   3. Construct the perpendicular bisector for segment AB. Label the new intersection
      point Y.
   4. Identify the midpoint of segment AB. Use a ruler to verify showing your work.


       E. Lesson 5: Angle Pairs

Read the definitions of adjacent, congruent, complementary angles, supplementary
angles, vertical angles and linear pairs.
The following is a diagram of angle pairs, where line l and x are parallel.




The following are angle pairs and their relationships for two parallel lines intersected by a
transversal.
  1,  2,  7 and  8 are exterior angles.
  3,  4,  5 and  6 are interior angles.
  1 and  8, and  2 and  7 are alternate interior angles.
  3 and  6, and  4 and  5 are alternate exterior angles.
  1 and  5,  2 and  6,  3 and  7, and  4 and  8 are corresponding angles.
  3 and  5, and  4 and  6 are consecutive interior angles.

Using the concepts above, complete the following activity for vertical angles and linear
pairs. Using Sketchpad software to answer, print out and complete the worksheet from
this activity.

                  Vertical Angles and Linear Pairs Using Sketchpad
Purpose:
To discover the relationships between pairs of vertical angles and between linear pairs.
Procedure:

   1. Select the Line tool and construct Line AB.
   2. Construct another line, making sure it intersects
      Line AB between Point A and Point B, and that
      Point C and Point D are on opposite sides of
      Line AB.
   3. Select the Arrow tool and highlight both lines.
   4. Select Construct, then Point of Intersection.
   5. Your figure should now resemble the figure shown above.
   6. We now want to measure some angles. To measure AEC, you select the Arrow
      tool, then select those three points in that order - A, E, C, - with the vertex letter
      of the angle in the middle. Once all three points are highlighted, select Measure,
      then Angle. Record this measurement in the space provided for that angle for
      intersection 1 in the chart on the next page.
   7. Repeat this procedure to measure the following angles: CEB, BED,
      and DEA. Record these measurements in the chart for intersection 1.
   8. Click and hold on either Point A, Point B, Point C, or Point D and drag it to a new
      location in order to create a new intersection. The only restriction to your
      movement is that Points A and B must remain on opposite sides of line CD and
      Points C and D must remain on opposite sides of Line AB, and that Point E
      remain between the other four points.
   9. Your measurements will change as you move the point. Record these new
      measurements for intersection 2. Repeat this process to complete the chart.

            Vertical Angles and Linear Pairs - Sketchpad Worksheet
Data:              m  AEC         m  CEB         m  BED      m  DEA
Intersection 1     ________       ________         ________     ________
Intersection 2     ________       ________         ________     ________
Intersection 3     ________       ________         ________     ________
Intersection 4     ________       ________         ________     ________
Intersection 5     ________       ________         ________     ________
Intersection 6     ________       ________         ________     ________
Intersection 7     ________       ________         ________     ________
Intersection 8     ________       ________         ________     ________
Intersection 9     ________       ________         ________     ________
Intersection 10    ________       ________         ________     ________
Intersection 11    ________       ________         ________     ________
Intersection 12    ________       ________         ________     ________
Intersection 13    ________       ________         ________     ________
Intersection 14    ________       ________         ________     ________
Intersection 15    ________       ________         ________     ________

Exploration:

   1. There are two pairs of vertical angles showing on the screen. Please identify these
      two pairs. ________________ and _________________
   2. What do you notice about each pair of vertical angles for each intersection?
      _______________________________________________________________
   3. There are four linear pairs of angles on the screen. Please identify all four.
   4.
               __________________ and ___________________
               __________________ and ___________________
               __________________ and ___________________
               __________________ and ___________________

   5. What do you notice about the sum of the measures for each linear pair at each
      new intersection?
      ________________________________________________________________

Conjectures:

   1. If two angles are vertical angles, then
      __________________________________________________________________
      _ ___________________________________________________________.
   2. If two angles form a linear pair, then
      __________________________________________________________________
      ____________________________________________________________.


       F. Lesson 6: Bisecting an Angle

After reading the definition for congruent angles, construct a congruent angle pair by
bisecting an angle.
Procedure:

   1. Draw a 50° angle using your protractor.
   2. Bisect the angle using your compass. Use your protractor to verify. (Each angle
      should now be 25°.)
   3. Draw a 128° angle using your protractor on a piece of wax paper.
   4. Bisect the angle by paper folding. Again, use your protractor to verify your work.
      (Each angle should now be 64°.)


       G. Lesson 7: Angle Pairs - An Origami Exercise

You will integrate the various types of angle pairs just studied with origami. Make either
an origami boat or whale. Remember to start with a square sheet of paper. You do not
have to use origami paper but it is better looking. A "squared" sheet of plain paper will
work fine.
Now unfold your origami shape and complete the following activity for angle pairs.
Remember to print out the worksheet and complete it.

               Geometry in Motion - Connections to Geometry Terms
      1. Unfold your origami shape.
      2. Sketch the unfolded shape below showing the creases made.
      3. Number the angles needed to identify the angle pairs below. Do NOT number
         ALL angles.
      4. Name 2 angle pairs for each of the following:
            a. complementary angles - _________________________________
            b. supplementary angles that are NOT adjacent -
                ________________________________
            c. vertical angles - ________________________________
            d. linear pair - ________________________________
            e. congruent angles formed by an angle bisector -
                __________________________________________________



II.       Unit 2: Identifying Transversals and Angles

          A. Unit 2 Introduction

The learner will identify transversals and the angles formed to prove and use theorems
related to angles formed by parallel lines and transversals, determine when lines are
parallel and solve life-related problems.

The learner will:

         find angle relationships such as vertical angles, linear pairs, and supplementary
          angles.
         identify relationships between and among points, lines, and planes, such as
          parallel and perpendicular lines.
         find the intersection of lines, planes, and solids.
         connect geometric diagrams with algebraic representations.
         integrate constructions such as segments and angles, and parallel lines.

          B. Lesson 1: Parallel Lines

Read the definition of parallel lines. Then, construct two parallel lines and a transversal
intersecting them. Label and measure the angles. You will need a straight edge and
protractor to do this. Your drawing should look similar to this:




(Label the picture as shown in order to do the following activity correctly!)
Now study your angle measures and see if you see a pattern/relationship in the numbers.
Then complete these sentences, use knowledge you learned from Chapter 1:

   1. Angle EGB and angle BGH form a ______________________ and their angle
      measures are ______________________.
   2. Angle EGB and angle GHD form _______________________________ angles
      and their angle measures are ___________________________.
   3. Angle BGH and angle GHD form _______________________________ angles
      and their measures are _______________________________.
   4. Angle BGH and DHF form ____________________________________ angles
      and their measures are _______________________________.
   5. Angle GHD and angle DHF form a ____________________________ and their
      angle measures are ______________________________.
   6. Name another angle pair that is the same type of angles as question 2.
      _____________ and _________________
      What do you think their measures are? ______________________
   7. Name another angle pair that is the same type of angles as question 3.
      ______________ and ___________________
      What do you think their measures are? ____________________
        C. Lesson 2: Transversal Angles

Next, you are going to use GSP to complete an activity in which you will construct
parallel lines yourself, measure angles, move points, and explain in words what are the
relationships between various types of angles. If you can print from GSP, please print out
your sketches for this activity. If you are working with the demo version then record your
sketches on paper. This is similar to lesson one but this time you are using Sketchpad
(GSP) and making more observations.

                              Pairs of Transversal Angles
Procedure:

   1.  Select the Line tool and construct a line AB.
   2.  Select the Point tool and construct a point C not on your line.
   3.  Select the Arrow tool, and then highlight both Line AB and Point C.
   4.  Select Construct, then Parallel Line.
   5.  You should see a line through Point C, which is parallel to Line AB.
   6.  Select the Point tool and construct a point D on Line AB between Point A and
       Point B.
   7. Select the Arrow tool, highlight both Point C and Point D, then select Construct,
       then Line.
   8. Select the Point tool and construct two points, E and F, on the line parallel to Line
       AB on opposite sides of Point C. Place E to the left of C and F to the right of C.
   9. We now want to measure some angles. To measure angle ADC, you select the
       Arrow tool, then select those three points in that order - A, D, C - with the vertex
       letter of the angle in the middle. Once all three points are highlighted, select
       Measure, then Angle. Record this measurement in the space provided for that
       angle for Transversal 1 in the chart below.
   10. Now measure angle BDC, angle ECD, and angle FCD in the same manner and
       record those measurements in the chart as well.
   11. Now click and hold on Point C or Point D and move it along the line in order to
       create a new construction. The only restriction to your movement is that Point D
       must remain between Point A and Point B, and that Point C must remain between
       Point E and Point F. Your measurements should change as you shift the point.
       Record these new measurements for Transversal 2.
   12. Continue the process until you have completed the chart.

Data:         m  ADC m  BDC m  ECD m  FCD
Transversal 1
Transversal 2
Transversal 3
Transversal 4
Transversal 5
Transversal 6
Transversal 7
Transversal 8
Transversal 9
Transversal 10
Transversal 11
Transversal 12
Transversal 13
Transversal 14
Transversal 15

Exploration #1:

   1. According to our definitions, what type of transversal angle pair are  ADC and
        FCD? _________
   2. What seems to be true about the measures of these two angles in all cases?
       __________________________________________________________________
       ____________
   3. According to our definitions, what type of transversal angle pair are  BDC and
        ECD? ________
   4. What seems to be true about the measures of these two angles in all cases?
       __________________________________________________________________
       ____________
   5. According to our definitions, what type of transversal angle pair are  ADC and
        ECD? _________
   6. For Transversal 1, what is the sum of these two angles?
       ______________________
   7. Does this sum hold true for the other transversals as well?
       ____________________
   8. According to our definitions, what type of transversal angle pair are  BDC and
        FCD? _________
   9. For Transversal 1, what is the sum of these two angles?
       ______________________
   10. Does this sum hold true for the other transversal as well?
       _____________________

Conjectures:

   1. When two parallel lines are cut by a transversal, the pairs of ________________
      _______________ angles formed are ______________________.
   2. When two parallel lines are cut by a transversal, the pairs of ______________
      ________________ angles formed are ____________________.

                     Pairs of Transversal Angles - The Sequel
Procedure:

   1.  Select the Line tool and construct a line AB.
   2.  Select the Point tool and construct a point C not on your line.
   3.  Select the Arrow tool, then highlight both Line AB and Point C.
   4.  Select Construct, then Parallel Line.
   5.  You should see a line through Point C, which is parallel to Line AB.
   6.  Select the Point Tool and construct a point D on Line AB between Point A and
       Point B.
   7. Select the Arrow tool, highlight both Point C and Point D, then select Construct,
       then Line.
   8. Select the Point tool and construct two points, E and F, on the line parallel to Line
       AB on opposite sides of Point C. Place E to the left of C and F to the right of C.
   9. Construct a point on Line CD above Line EF (Point G) and a point on Line CD
       below Line AB (Point H).
   10. We now want to measure some angles. To measure ADH, you select the Arrow
       tool, then select those three points in that order - A, D, H - with the vertex letter of
       the angle in the middle. Once all three points are highlighted, select Measure, then
       Angle. Record this measurement in the space provided for that angle for
       Transversal 1 in the chart below.
   11. Now measure the following angles: BDH, ADC, BDC, ECD,
         FCD, ECG, and FCG. Record these measurements in the chart as well.
   12. Now, click and hold on Point C or Point D and move it along its line to create a
       new construction. The only restriction to your movement is that point D must stay
       between point A and point B and point C must stay between point E and point F.
       Your measurements should change as you shift the point. Record these new
       measurements for Transversal 2.
   13. Continue this process until you have completed the chart.

              m       m        m        m        m        m        m        m
Data:
              ADH      BDH       ADC       BDC       ECD       FCD       ECG       FCG
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal
Transversal

Exploration #2:

   1. According to our definitions, what type of transversal pair are each of the
      following pairs of angles:

                ADH and  ECD? ____________________________
                BDH and  FCD? ____________________________
                ADC and  ECG? ____________________________
                BDC and  FCG? ____________________________

       2. In comparing each pair of angles for each transversal, what would seem to be
       true of the measures of each pair?
       _________________________________________________________________

Conjecture:
When two parallel lines are cut by a transversal, the pairs of _______________________
angles are __________________________________.


       D. Lesson 3: Angles Used in Parallel Lines.

Read the definitions of parallel lines, same side interior angles, alternate exterior angles,
alternate interior angles, corresponding angles, vertical angles, linear pairs, congruent and
supplementary.
Rules to know:

      If two parallel lines are cut by a transversal, then the alternate exterior angles are
       congruent.
      If two parallel lines are cut by a transversal, then the alternate interior angles are
       congruent.
      If two parallel lines are cut by a transversal, then the same side interior angles are
       supplementary.
      If two parallel lines are cut by a transversal, then the corresponding angles have
       the same measure.
      If two coplanar lines are cut by a transversal and the corresponding angles have
       the same measure, then the lines are parallel.
Now complete these questions:

Identify the pairs of angles.
   1.  9 and  8 _____________
   2.  12 and  3 ___________
   3.  11 and  8 ____________
   4.  11 and  7 ____________
   5.  9 and  5 _____________
   6.  1 and  11 ____________
   7.  3 and  4 _____________


State which lines are parallel and why.

   8.  1  14 _________, _______________________
   9. m  9 = 120o,
       m  11 = 120o __________, ___________________
   10. m  8 = 50o,
       m  6 = 130o _________, ___________________
   11. m  12 = m  15 ________, ___________________
   12.  7  1 ________, ________________________
   13. m  3 = 132o,
       m  2 = 132o _________, ___________________




Given: Line XY || line GH.

   14. If m  YWQ = 53o, find m  XWQ, m  GJK, m  QJH, and m  RWY. _____,
       ______, ______, ________
   15. * If m  YWQ = 4x . 10 and m  GJK= 2x + 36, find m  YWQ and m  QJH.
       _______, ________
   16. * If m  XWQ = 5x and m  QJH = x + 24, find m  YWQ & m  QJG.
       _______, ________ (Use for 14-16.)
* Show work.




Complete.

   20. State a practical, "real-world" example of skew lines/segments.
       __________________________________________________________________
   21. State a practical, ―real-world‖ example of essential parallel lines/segments.
       __________________________________________________________________
   22. What is similar about parallel and skew lines?
       _____________________________
       What is different?
       ____________________________________________________


       E. Lesson 4: Slopes of Perpendicular and Parallel Lines

Read the definitions for parallel lines, perpendicular lines, slope, reciprocal, and inverse
reciprocal.
Things to know about parallel and perpendicular lines:

      Two non-vertical lines are parallel if and only if they have the same slope.
      Lines that are parallel have the same slope.
      Lines that are perpendicular have slopes that are inverse reciprocals.
Example: If line l has a slope of 10, what is the slope of line x if they are perpendicular?
      Since the lines are perpendicular, we need to find the inverse reciprocal of the
                                                                            1
      slope of line l. Line l has a slope of 10. The reciprocal of 10 is       and the inverse
                                                                           10
                       1       1                               1
      (opposite) of       is - . So, the slope of line x is - .
                      10      10                              10

Finish this lesson by answering these questions.
Given each pair of lines' slopes, state whether the lines are parallel, perpendicular, or
neither.

       1.   2, 2
       2.   -2, -1/2
       3.   -5/6, 6/5
       4.   -7, 1/7

State whether the pairs of lines determine by each set of points is parallel, perpendicular,
or neither. Remember to first find the slope for each line. Finding the slope of a line is an
Algebra I concept that you should have already had. If you have forgotten how to
calculate slope, then go to an Algebra I course and review how to calculate slopes of lines
given two points.

       5. Line AB and line CD: A (1, 2); B (2, 3) and C (8, -1); D (7, -2).
       6. Line EF and line GH: E (3, 8); F (-2, -9) and G (-2, 0); H (9, -5).
       7. Line ST and line XY: S (-5, -2); T (-4, 1) and X (7, 6); Y (8, 3).

III.        Unit 3: Triangles

            A. Unit 3 Introduction

The learner will investigate the types of relationships of triangles using the properties that
relate to the segments and angles of triangles including inequalities, perpendicular
bisectors, altitudes, and medians, determine missing measures using diagrams and
appropriately apply theorems/corollaries for equilateral, isosceles, scalene triangles and
interior/exterior angle properties to solve problems.

The learner will:

           connect geometric diagrams with algebraic representations.
           integrate constructions such as segment bisectors, perpendiculars, and polygons.
           describe, draw, and construct two-dimensional figures.
           use angle and side relationships such as triangle inequalities, isosceles and
            equilateral triangle properties, altitude, and median.

            B. Lesson 1: Angles in Triangles
Read the definitions for acute angle, obtuse angle, right angle, acute triangle, right
triangle, and obtuse triangle.

You are now to create a collage of the different types of triangles that are in your world.
You will need at least three pictures for each of these triangles: acute triangle, right
triangle, and obtuse triangle. These pictures can come from the Internet or magazines or
your own photographs. After you have located your pictures, then display them on a
poster board, appropriately labeled.

       C. Lesson 2: Measuring Angles

This lesson will review measuring angles with a protractor. To measure an angle with a
protractor, you need to place the measuring line with the center, usually a hole, on the
vertex and a line of the angle. Then, look at the angle measurements on the protractor to
read the angle measurement. Keep in mind that an acute angle is less than 90° and that an
obtuse angle is greater than 90°.

Now, measure the following angles with your protractor:




1.                               2.




3.                                      4.

5.

6.


       D. Lesson 3: Interior Angles

This lesson will introduce you to one of the most used theorems in geometry.
Theorem: The sum of the interior angles in any triangle is 180°.
Examples: Using the following picture complete the problems.




   1.  ABC = 48°,  ACB = 36°, find  BAC
      We know that the angles inside of a triangle equal 180°. So,
       ABC+  ACB+  BAC = 180°
       48°+36°+  BAC = 180°
       84°+  BAC = 180°
      -84°         -84°
        BAC = 96°

   2.  ACB = 2x,  BAC = x + 10,  ABC = 3x + 8, find x,  ACB,  BAC and
       ABC
      We know that the angles inside of a triangle equal 180°. So,
       ABC+  ACB+  BAC = 180
      2x + (x + 10) + (3x + 8) = 180 (Combine like terms)
      6x + 18 = 180
           -18 -18
      6x = 162
       x = 27°
       So,  ACB = 2x = 2(27°) = 54°
           BAC = x + 10 = 27° + 10° = 37°
           ABC = 3x + 8 = 3(27°) + 8° = 81° + 8° = 89°
          And, 54° + 37° + 89° = 180°

   3.  ABC = 46°,  ACB = 42°, find  BAC and  ABD
      We know that the angles inside of a triangle equal 180°. So,
       ABC+  ACB+  BAC = 180°
       46°+42°+  BAC = 180°
       88°+  BAC = 180°
      -88°         -88°
       BAC = 92°
      And we know that  ABD +  ABC = 180°.
        ABD + 46° = 180°
              -46° -46°
        BAC = 134°

Last, practice using this new theorem, combined with a little algebra skills, on the
following problems:
Given: triangle ABC.

   1. m  A= 56 ½°, m  B= 29°, m  C= __________.
   2. m  A= 3x, m  B= x + 10, m  C= 2x + 14, x= ________,
      m  A= ___________, m  B= _____________, m  C= _____________.




Use the drawing to the below to find the indicated angle measures' sums. State a reason
for each.

   8. m  2 + m  4 + m  6= ______; ____________________
   9. m  6 + m  1= ________; ______________________
   10. m  1 + m  3 + m  5 = _________; _________________
       E. Lesson 4: Sides of a Triangle

We have looked at the types of triangles according to their angles and have worked with
the angle measures of a triangle. Now, we are going to explore relationships that exist for
the sides of a triangle and how they relate to the angles of a triangle.
Rules for equilateral triangles:
     If a triangle is equilateral, then it is equiangular.
     If two angles of a triangle are congruent, they are the base angles of an isosceles
        triangle.
     If a triangle is equiangular, then it is equilateral.

Now try working these problems:
Given: Triangle RST is isosceles with  T the vertex angle. Find:

   1. m  R = 64o , m  S = _________, m  T = _____________.
   2. m  T = 64o , m  S = _________, m  R = _____________.
   3. m  T = x + 10, m  R = 3x + 15, m  T = __________, m  R = ___________,
      and m  S = __________.
   4. RT = 3x - 1, TS = x + 9, RS = 2x + 1, RS = ____________.

       F. Lesson 5: Triangle Inequality Theorem

We have been working with triangles throughout this unit. Now let's explore this
question: Can a triangle be formed given any three segment lengths? Your first response
is probably ―yes‖. If you have a piece of uncooked spaghetti at your house, go get two
pieces. Break the first piece into three relatively the same length pieces and see if you can
form a triangle. Remember to join the pieces end-to-end. It should have worked. Now
take the second piece of spaghetti and break two very small pieces off one end of it.
(About 1 inch each.) The long piece should still be 4-5 inches long. Now try making a
triangle as you just did. Any problems this time? Did your pieces of spaghetti look
something like this?


Tape the two sets of spaghetti pieces on a sheet of paper to be turned into your teacher.
Now, you are going to read about the Triangle Inequality Theorem.
Triangle Inequality Theorem: The sum of the measures of any two sides of any triangle is
greater than the measure of the third side. In other words, you can pick any two sides
measure and when they are added together the sum will be greater than the measure of
the third side.
Example: Can the following lengths form a triangle?
    1. 8, 6, 2
        8 + 6 = 14 > 2 ok
        8 + 2 = 10 > 6 ok
        6 + 2 = 8 not >8 fails
        So, no, can’t form a triangle

   2. 3, 4, 5
      3 + 4 = 7 > 5 ok
      3 + 5 = 8 > 5 ok
      4 + 5 = 9 > 3 ok
      So, yes, can form a triangle

Now practice the Triangle Inequality Theorem.
Can a triangle be formed with the following lengths? Explain each answer.

   1.   2", 3", 5"
   2.   4 cm, 1 cm, 2 cm
   3.   6', 8', 1'
   4.   5 m, 5 m, 5 m
   5.   2x, 3x, 4x

        G. Lesson 6: Altitudes, Medians, and Bisectors

Read the definitions for altitudes, medians, angle bisectors, and perpendicular bisectors of
triangles.
Pictures of properties for altitudes, medians, and perpendicular bisectors.
Median




Altitude
   In an acute triangle, all of the altitudes are inside the triangle.




   In a right triangle, one of the altitudes is inside the triangle and the other two are
    the legs of the triangle.
      In an obtuse triangle, one of the altitudes is inside the triangle and the other two
       are outside the triangle.




Perpendicular bisector




Use the information you learned and your algebra skills to answer the following
questions:

1. Segment AD is an altitude.
m  ADB = 5x - 10.
m  ABD = 2x + 5
Find x, m  ABD, and m  BAD.
2. Segment AD is an angle bisector.
m  BAD = 3x - 5
m  DAC = x + 29
Find x, m  BAD, and
m  BAC.

3. Segment AD is median.
BD = 5x + 2
DC = 2x + 8
Find x, BD, and BC.

4. Segment ED is a perpendicular
bisector.
BD = 3x - 3
DC = x + 2
Find x, BD, BC, and m  EDC.
5. Write a paragraph for each of the four types of special segments that you have
explored in this lesson describing their placement in reference to the triangle as the
triangle changes from acute to right to obtuse.

        H. Lesson 7: Euler Line

This last lesson will have you exploring the very famous Euler line. You will be using
some of the concepts from the last lesson in a GSP activity and also using your wax
paper. When the second worksheet refers to a "patty paper," you use about a 4 inch
square piece of wax paper. After you complete the first activity, be sure to print out the
completed worksheet and the sketch from GSP (or a drawing on graph paper if using the
demo version of GSP.)

               SKETCHPAD ACTIVITY - THE EULER LINE
Name: _______________________________________
PROCEDURE:

   1. Select the Point Tool and create three points.
   2. Select the Arrow Tool and highlight all three points.
   3. Construct Segment to form a triangle.
   4. While the segments are still highlighted, Construct Point at Midpoint.
   5. Individually, highlight each midpoint and the segment on which it lies, then
      Construct Perpendicular Line. Do this for all three sides of the triangle.
   6. You have now created the three ________________________
      ______________________ of your triangle.
   7. Highlight any two of these ________________________
      _______________________ and Construct Point as Intersection. This should
      be Point G. This point is named the __________________.
   8. Do all three of these ____________________ _____________________ intersect
       in the same point? _______
   9. Individually, highlight each side of the triangle and the vertex opposite that side,
       then Construct Perpendicular Line. Do this for all three sides of the triangle.
   10. You have now constructed the three _________________________ of the
       triangle.
   11. Highlight any two of these _________________________ and Construct Point
       at Intersection. This should be Point H. This point is named the
       ____________________________.
   12. Change the Point Tool to the Line Tool.
   13. Select the Arrow Tool, highlight Points G and H, then Construct Line.
   14. Individually, construct the segment from each vertex to the midpoint of the
       opposite side. Do this for all three sides of the triangle.
   15. You have now created the three ________________________ of the triangle.
   16. Highlight any two of these ______________________ and Construct Point at
       Intersection. This should be Point I. The name of this point is
       _________________________.
   17. Is Point I on Line GH? ________________


                EULER LINE ACTIVITY WITH PATTY PAPERS
Procedure:

   1. Trace the three basic triangles—acute, right, and obtuse—onto patty papers. The
      students will need to trace three of each type of triangle on three separate patty
      papers.
   2. On one acute triangle construct the perpendicular bisector for each side. To
      construct a perpendicular bisector with the patty paper, you must fold the paper so
      that one side lies on top of itself until one endpoint is lying on top of the other
      endpoint. Then crease the paper along this fold. Trace along each of the folds to
      display the three perpendicular bisectors. When all three perpendicular bisectors
      have been constructed, darken the point of intersection of these three lines. You
      have constructed the circumcenter of the triangle.
   3. On a second acute triangle, construct the three altitudes of the triangle. Perhaps
      the best way to do this is to slide an index card along one side of the triangle until
      the vertical edge lines up with the opposite vertex while the horizontal edge is
      aligned with the side of the triangle. Trace or fold along the vertical edge of the
      index card to construct the altitude to that side of the triangle. After all three
      altitudes have been constructed, darken the point of intersection of the three
      altitudes. You have constructed the orthocenter of the triangle.
   4. On the third acute triangle construct the three medians of the triangle. Use the
      same basic procedure discussed above for the perpendicular bisectors in order to
      locate the midpoints of the three sides of the triangle. Then use a straightedge to
      connect each vertex to its opposite midpoint. After all three medians have been
      constructed, darken the point of intersection of the three medians. You have
      constructed the centroid of the triangle.
      5. Now you can lay the three patty papers on top of each other so that all three
         triangles are aligned with each other. You may then use a straightedge to see that
         these three points of intersection—the circumcenter, the orthocenter, and the
         median—all lie on the same line—the Euler line.

Follow the same procedure for the other two triangles to demonstrate that this
construction applies to all three types of triangles.

IV.       Unit 4: Pythagorean Theorem
          A. Unit 4 Introduction

The learner will utilize the Pythagorean Theorem, its converse, special right triangles,
and the trigonometric functions sine, cosine, and tangent to find missing measures in
right triangles, determine the type of triangle according to its angles, and solve problems.

The learner will:

         connect geometric diagrams with algebraic representations.
         use Pythagorean theorem and its converse.
         use right triangle relationships such as trigonometric ratios (45-45-90 and 30-60-
          90 triangles).

          B. Lesson 1: The Pythagorean Theorem

Read the definitions of the Pythagorean Theorem and Pythagorean Triples.

          C. Lesson 2: Using the Pythagorean Theorem

Now, finally we are going to work out problems using the Pythagorean Theorem

Examples: Using the picture and given information we will solve the following problems.




1. Given a = 4 and b = 3, find c.               2. Given a = 6 and c = 10, find b.
               a2 + b2 = c2                               a2 + b2 = c2
               (4)2 + (3)2 = c2                           (6)2 + b2 = (10)2
               16 + 9 = c2                                36 + b2 = 100
                25 = c2                                   -36         -36
                 25 =         c2                            b2 = 64
                5=c                                         b=8

Problems:

   1.   Given a = 9 and b = 12, find c.
   2.   Given a = 5 and c = 12, find b.
   3.   Given b = 13 and c = 3, find a.
   4.   Given a = 3 and b = 9, find c.


        D. Lesson 3: Converse of the Pythagorean Theorem

This lesson will allow you to see how the Converse of the Pythagorean Theorem can be
used as well as more practice using the Triangle Inequality Theorem from the last unit.
Read the definition of the Converse of the Pythagorean Theorem.

The following rules apply to triangles and the Converse of the Pythagorean Theorem.
    In any triangle




        a+b>c
        b+c>a
        a+c>b
       In any triangle, a < b, if and only if  A <  B.




       All of a triangles sides are of equal length if and only if all of its angles are equal.




       In a right triangle
        c2 = a2 + b2
       In an acute triangle



        c2 < a2 + b2
       In an obtuse triangle




       c2 > a2 + b2
Now try answering these questions.
State which types of triangles according to angles and sides would occur with each set of
segments given. Explain your answers.

   1.   5", 7", 10"
   2.   8 cm, 8 cm, 12 cm
   3.   3 m, 3 m, 4 m
   4.   6', 8', 10'
   5.   x, x, x

        E. Lesson 4: Special Right Triangles

Now we are going to learn "Shortcut rules" for certain special right triangles. These
triangles will either have angle measures of 45o, 45o, and 90o; or 30o, 60o and 90o.

In a 45o - 45o - 90o , an isosceles triangle, the length of the hypotenuse is equal to the
length of a leg multiplied by 2 .




In a 30o - 60o - 90o , the length of the hypotenuse is two times that of the leg opposite
the 30o angle. The length of the other leg is 3 times that of the leg opposite the 30o
angle.




Examples:
1. Given a 45o - 45o - 90o with  A = 90o and AB = 4cm. Find AC and BC.
       We know that AC = AB (legs of the triangle). So, AC = 4cm.
       BC = AC( 2 ) = AB( 2 ) = 4 2 cm
2. Given a 30o - 60o - 90o with  B = 60o,  C = 30o and BC = 6cm. Find AB and AC.
       We know that BC in the hypotenuse by the given information, so BC = 2x = 6cm.
                          2x   6cm
       Now solve for x:      =       x = 3 cm.
                          2     2
       So, AB = 3cm. AC = x( 3 ) = 3 3 cm.

Now see if you can work these problems.

Complete using the "quick" rules for 45o-45o-90o and 30o-60o -90o triangles.
Given: m  B = m  C = 45o.

   1.    AC = 8", find AB & BC.
   2.    AB = 6.3 cm, find AC & BC.
   3.    BC = 17 m, find AB & AC.
   4.    BC = 2 2 ft, find AB & AC.

Given: m  C = 60o, m  B = 30 o.

   5.    AC = 3", find AB & BC.
   6.    BC = 29 m, find AC & AB.
   7.    AB = 5.3 cm, find AC & BC.
   8.    BC = 7.9 m, find AC & AB.

         F. Lesson 5: Right Triangle Trigonometry

This lesson will introduce you to right-triangle trigonometry. If you take a pre-calculus
class at some time, you will learn more about trigonometry, specifically circular
trigonometry. In geometry, though, we just work with right triangles. Once you learn the
three basic trig ratios and how to enter the information into your scientific calculator, you
should find trig a very easy concept to use. One reminder when using your scientific
calculator, it should always be in the "degree" mode. There are other units of measure for
angles such as radians and gradians, but we will only measure angles in degrees in this
course. First, read the definitions of sine, cosine, and tangent, then review the formulas.




           opposite      O
sine x =              =
         hypotenuse      H
             adjacent     A
cosine x =              =
            hypotenuse    H
             opposite   O
tangent x =           =
             adjacent   A




                            c            c                b
For this image, sin (C) =     , tan (C) = , and cos (C) =
                            a            b                a
Example:




Find a and b.
           a     a                                a     8.66
sin (A) = =                            tan (A) =     =
           c 10                                   b       b
             a                                     8.66
sin (60) =                             tan (60) =
           10                                        b
                   a                                    8.66
(sin (60))(10) =     (10)              (tan (60))(b) =       (b)
                  10                                      b
(sin (60))(10) = a                     (tan (60))(b) = 8.66
                                      (tan (60))(b)     8.66
(0.866)(10) = a                                     =
                                         tan (60)     tan(60)
8.66 = a                              b=5

To find the degrees of a reference angle you will need to use the following function keys
on your calculator:
       sin-1, to use when you find sine
       cos-1, to use when you find cosine
       tan-1, to use when you find tangent

Examples:
   1.




        Find  C and  B.
                    1                              1                     1
        sin (C) =       = 0.707     or tan (C) = = 1 or cos (C) =            = 0.707
                     2                             1                      2
        sin-1 (0.707) = 45 o            tan-1 (1) = 45 o     cos-1 (0.707) = 45 o
        So,  C = 45 (you need only use any one of the above three)
                       o

        Since  C = 45 o and  A = 90 o, then  B = 45 o.
   2.




        Given a = 11.1 and c = 16.7. Find  A and  B.
        For  A: a is the opposite and c is the hypotenuse.
        For  B: a is the adjacent and c is the hypotenuse.
        So, for  A use sine and  B use cosine formulas.
                   11.1                                   11.1
        sin (A) =       = 0.66467              cos (B) =       = 0.66467
                   16.7                                   16.7
        sin-1 (0.66467) = 41.7 o               cos-1 (0.66467) = 48.3 o
        So,  C = 47.1 and  B = 48.3 .
                        o                 o



Now let's see if you can work problems on your own. You can, and should, use your
scientific calculator.
State the correct ratio of sides for each.

     1. Sin =
     2. Cos =
     3. Tan =

State the trig equation and solve for each. Each triangle is a right triangle.




Given:    ABC with sides a, b, and c with m C = 90o. State the trig equation and solve.

     8. m A = 63o, b = 28 ft, a = _________
     9. m B = 29o, a = 13.2 cm, c = ________
     10. b = 12.9 cm, c = 18.7 cm, m A = _________o; m B = _________o

V.       Unit 5: Quadrilaterals
         A. Unit 5 Introduction

The learner will recognize special quadrilaterals and the appropriate properties associated
with each quadrilateral and utilize these properties and drawings to apply formulas of
area, surface area, and volume for a variety of geometric shapes and solids to calculate
these measurements.

The learner will:

        describe, draw, and construct two-dimensional and three-dimensional figures.
        use properties of quadrilaterals such as classification.
        use properties of other polygons.
        use relationships among one-, two-, and three-dimensional measures.
        use perimeter, circumference, and area of planar regions to determine volume and
         surface area of solids.
        use properties of circles.

         B. Lesson 1: Basic Polygons

In this lesson, you will practice naming polygons as well as drawing polygons. Read the
definitions for concave, convex, equilateral triangle, heptagon, hexagon, isosceles
trapezoid, isosceles triangle, octagon, parallelogram, pentagon, rectangle, right triangle,
scalene, and trapezoid, then complete the following assignment. You will need coloring
pencils, a ruler, and protractor for the drawing part of this lesson.
Assignment:
Draw each of the following polygons: equilateral triangle, heptagon, hexagon, isosceles
trapezoid, isosceles triangle (non-equilateral), octagon, parallelogram (not a rectangle or
rhombus), pentagon, rectangle, right triangle, scalene triangle, and trapezoid (non-
isosceles). Also, write a paragraph for each graph explaining how you created it.

       C. Lesson 2: Special Quadrilaterals

Read the definitions for congruent, diagonal, kite, parallelogram, quadrilateral, rectangle,
rhombus, square, and trapezoid. You will now explore some specific polygons and
special quadrilaterals. (Note: Not every quadrilateral has parallel sides or equal sides!)

The following are Special Quadrilaterals:
       Kite (can be a parallelogram because if two pairs of sides are equal, then it is a
             rhombus or if the angles are all equal then it is a square)
       Parallelogram
       Rectangle (also a parallelogram, since it has two pairs of parallel sides)
       Rhombus (not always a rectangle because it does not have four right angles and
                  not all four sides of a rectangle have to be equal)
       Square (also a rectangle and a parallelogram)
       Trapezoid (not a parallelogram because it has only one pair of parallel sides)

The Four Theorems for Parallelograms:
    The diagonal of any parallelogram forms two congruent triangles.
    Both pairs of opposite sides in a parallelogram are congruent.
    Both pairs of opposite angles in a parallelogram are congruent.
    The diagonals of any parallelogram bisect each other.


       Ex.

Three Theorems to show a Quadrilateral is a Parallelogram
    If both pairs of opposite sides of a quadrilateral are congruent, then the
       quadrilateral is a parallelogram.
    If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a
       parallelogram.
    If one pair of the opposite sides of a quadrilateral are both parallel and congruent,
       then the quadrilateral is a parallelogram.

Example:
Rectangle:
Find  ABC, CD, BC, BD, and AC.
Since the image is a rectangle,  ABC = 90 o, CD = AB = 3‖, and BC = AD = 8‖. Now,
find BD and AC. We do know that BD = AC by definition of a rectangle, so solve for
only one segment and you’ll know both segment lengths.




Use the Pythagorean Theorem to find BD, also labeled as c.
a2 + b2 = c2
(8)2 + (3)2 = c2
64 + 9 = c2
73 = c2
 73 = c 2
 73 = c

Now let's practice working some problems applying the properties of special
quadrilaterals.
Find each indicated length or angle measure.
          D. Lesson 3: Computing Area

You to refresh your memory about computing area of two-dimensional shapes to help
you with finding the surface area of three-dimensional figures in the next lesson. Most
students will have had area and perimeter of two-dimensional figures usually in the
seventh and/or the eighth grades. That's been some time ago. So, let's review. Start by
reading the definitions for area and perimeter. Then, review the following formulas.

Area and perimeter formulas that you should need:

Circle:




        Area =  r2                  Perimeter = 2  r
Ellipse:




                                                         (r1 ) 2  (r2 ) 2
          Area =  r1r2              Perimeter = 2 
                                                                 2
Equilateral Triangle:




                    3 2
          Area =     (a )            Perimeter = 3a
                   4
Parallelogram:




      Area = bh                      Perimeter = 2a + 2b
Rectangle:
     Area = ab                        Perimeter = 2a + 2b
Rhombus:




                1
       Area =     d1d2                Perimeter = a + b + c + d
                2
Square:




      Area = s2                       Perimeter = 4s
Trapezoid:




                1
       Area =     h(b1 + b2)                 Perimeter = a1 + b1 + a2 + b2
                2
Triangle:




                1
       Area =     bh                  Perimeter = a + b + c
                2

For solid, or 3-d, figures use the following formulas for total area (T) and volume (V).
Cone:
                                          1 2
       T =  rl +  r2               V=     rh
                                          3
Cylinder:




      T = 2  rh + 2  r2            V =  r2h
Pyramid:




           1                              1
       T=    PL                      V = Bh
           2                              3
       Where, B is the area of the base and P is the perimeter.
       For T we are using a regular pyramid which has a base that is a regular polygon
       and with lateral faces that are all congruent isosceles triangles.
Right Prism:




       T = 2B + Ph                 V = Bh
       where, B = lw and P = 2l + 2w
Sphere:




                                          4
       T = 4  r2                    V=      r3
                                          3

Example:




Find the volume.
V = lwh
V = (7.5 cm)(3 cm)(2cm) = 45 cm3
Assignment:
Find the area and perimeter from the given information.
   1. The side of a square is 2‖.
   2. The length of a rectangle is 4 cm and the width is 3 cm.
   3. The base of a triangle is 6‖, its height is 4‖, and both legs are 5‖.
   4. A circle has a diameter of 5 cm.

Find the volume for each solid.
   1.




   2.
   3.




   4.




   5.




        E. Lesson 4: Three-Dimensional Figures/Polyhedras

Now we are going to switch to polyhedras. Begin by reading the definitions for cube,
dodecahedron, icosahedrons, octahedron, polyhedra, tetrahedron, and vertices.
Vertices and edges in the polyhedras we will be looking at:
Tetrahedron - has four vertices and six edges.
Octahedron - has six vertices and twelve edges.
Cube – has eight vertices and twelve edges.
Icosahedrons - has twelve vertices and thirty edges.
Dodecahedron - has twenty vertices and thirty edges.

Assignment:
For this assignment you will need scissors, glue or tape, paper, and the given pictures,
because you are going to construct the five regular polyhedra that you explored earlier.
Print out the following four pictures of the regular polyhedra and make one yourself for a
cube, then construct them using your scissors and either tape or glue. A nice way to
display your solids is to create a mobile. To create a mobile, you will need some type of
frame. A wire clothes hanger works very well. (You will need wire snips to cut it.) You
will also need some type of string to hang the solids. Either nylon thread or fishing line
works well. Experiment with making your mobile having different levels by using more
than one piece of clothes hanger, hanging two solids on one level and three solids on the
other level. Work with the solids to see what balances the best.
VI.       Unit 6: Congruent Polygons and Transformations
          A. Unit 6 Introduction

The learner will recognize the conditions needed to prove polygons congruent and/or
similar and corresponding parts. Then the learner will use the properties of
transformations in connection with congruency and similarity to draw images of figures
as reflections, translations, rotations, and dilations.

The learner will:

         connect geometric diagrams with algebraic representations.
         prove triangles and other polygons congruent and similar, and explore
          corresponding parts relationships.
         use reflections, translations, rotations, and dilations.

          B. Lesson 1: Introduction to Congruent Triangles

There are three ways to prove that two triangles are congruent: ASA (Angle-Side-Angle),
SAS (Side-Angle-Side), and SSS (Side-Side-Side). Read the definitions for congruent,
congruent angles, ASA, SAS, and SSS. For each of them you always get only ONE,
unique triangle. Some people will try to prove congruency by SSA (Side-Side-Angle).
This is not possible because with SSA you get two possible triangles with the same
measures but theses two triangles are not congruent. Therefore it cannot be used to
guarantee congruency for triangles.

          C. Lesson 2: Similar Triangles

Now we are going to learn about similar triangles. Read the following definitions: ASA
part 2, SAS part 2, SSS part 2, and similar polygons. Hopefully, you will quickly see the
difference between congruent and similar shapes.
The three Similarity Rules for Triangles are:
     angle-angle similarity
     side-angle-side similarity
     side-side-side similarity
Example: Given two similar triangles, find the missing values of a, y, and the measure of
 A..
1st: Write proportions for the corresponding sides of the triangles.
        15 y                           15 7.5
                               and       
        12 8                           12 a
2nd: Use cross products to solve.
        15 y                           15 7.5
                               and       
        12 8                           12 a
        (15)(8) = (y)(12)       and    (15)(a) = (7.5)(12)
        120 = 12y               and    15a = 90
        120 12y                        15a 90
                               and         
         12    12                       15 15
        10 = y                  and    a=6
3rd: Solve for  A.
         X and  A are corresponding angles and by definition of similar polygons
         A =  X = 44

Assignment:
   1. Given two similar triangles find c, d, and  F.




   For 2 – 6 state if the triangles are similar and by which rule.
   2.
3.




4.




5.




6.
       D. Lesson 3: Similarity and Area, Volume, and Scale

This lesson will have you explore how similarity relates to other geometric concepts such
as area, volume, and scale drawings. From previous lessons, we know that if two shapes
are similar, their corresponding sides have the same ratio and their corresponding angles
are equal. But what you may not know is that if two shapes are similar, then their lengths,
area, and volumes also have the same ratio. This means that for similar shapes:
                                  a
     Ratio of lengths = a:b or
                                  b
                                  a2
     Ratio of areas = a2:b2 or 2
                                  b
                                     a3
     Ratio of volumes = a :b or 3
                             3 3
                                     b
Examples:
    1. Given that the two shapes are similar find the missing variable.




   The ratios of the areas is 36:81
   The ratios of the lengths is 4:x
   To solve and compare the ratios we need them to all be to the first powers.
   a2 = 36 and b2 = 81, to get just a and b take the square root of both sides
   a2 = 36        a 2  36     a=6
   b2 = 81  b2  81  b = 9
   Now we have everything to the first power so that we can compare ratio to ratio.
   6 4
       (the single power ratios of area to the single power ratios of length)
   9 x
   To solve use cross products.
   6 4                                               6x 36
           (6)(x) = (9)(4)  6x = 36                        x=6
   9 x                                                6   6
   2. Given that the two shapes are similar what is the ratio of their areas?




   The ratios of the volumes is 8:64
   To solve and compare the ratios we need them to all be to the first powers.
   a3 = 8 and b3 = 64, to get just a and b take the cube root of both sides
   a3 = 8      3
                    a3  3 8    a=2
   b = 64  b  3 64  b = 4
    3               3   3

   Now we have everything to the first power. The ratio of the lengths is now 2:4. To get
   the ratios of the areas we just need to square a and b, since all sides are equal this is
   possible. a2 = 22 = 4 and b2 = 42 = 16
   So, the ratio of the areas is 4:16.

Assignment: Solve for the variables.
   1.




   2.
   3.




        E. Lesson 4: Transformations

This lesson will explore transformations and their connections to congruency and
similarity. Read the definitions for reflection, rotation, line of symmetry, transformation,
and translation.
Examples:
Rotation: Rotate A 90 .




Translation: Translate A to the left and down.




Reflection: Reflect B.
Symmetry: Show where the given triangle is symmetric by its line of symmetry.




Assignment:
What letters of the alphabet have symmetry? Show their symmetry. (Hint: Some have
more than one line of symmetry!)


Answer Key for Assignments in Geometry:
Unit 1 Lesson Answers

Unit 1, Lesson 1
       TD = 23, DR = 15
Unit 1, Lesson 2

   a. AB =     13, M: (5.5, 1)
   b. AB =     10, M: (-7.5, -1.5)
   c. AB =     65, M: (1.5, -1)

Unit 1, Lesson 5
Exploration

   1.  AED and CEB; AEC and DEB
   2. They measure the same (they are congruent)
   3.  AED and DEB; DEB and BEC; BEC and                    CEA;
       CEA and AED
   4. The sum is 180 degrees.

Conjectures

   1. their measures are equal (the angles are congruent)
   2. the sum of their measures is 180 degrees
Unit 2 Lesson Answers

Unit 2, Lesson 1

   1.   straight line, 180o
   2.   corresponding angles, varies
   3.   consecutive interior angles, 180o
   4.   corresponding angles, varies
   5.   straight line, 180o
   6.    EGA and  GHC,  AGH and  CHF, or  BGH and  DHF; varies
   7.    AGH and  GHC, 180o

Unit 2, Lesson 2

Exploration #1

   1. alternate interior angles
   2. they measure the same (are congruent)
   3. alternate interior angles
   4. they measure the same (are congruent)
   5. same side interior angles
   6. 180o
   7. yes
   8. same side interior angles
   9. 180o
   10. yes

Conjecture

   1. alternate interior angles; congruent
   2. same side interior angles; supplementary

Exploration #2

   1. all pairs are corresponding angles
   2. they are congruent (measure the same)

Conjecture
      Corresponding angles are congruent (have the same measure).

Unit 2, Lesson 3

   1.   same side interior angles
   2.   alternate exterior angles
   3.   alternate interior angles
   4.   corresponding angles
   5. vertical angles
   6. linear pairs (supplementary angles)
   7. same side interior angles
   8. v and w; alternate exterior angles are congruent
   9. g and h; corresponding angles are congruent
   10. v and w; same side interior angles are supplementary
   11. v and w; corresponding angles are congruent
   12. g and h; corresponding angles are congruent
   13. v and w; alternate interior angles are congruent
   14. 127o, 53o, 53o, 127o
   15. 82o, 82o
   16. 50o, 130o
   17. x = 6, y = 10
   18. x = 147, y = 33, z = 10
   19. x = 59, y = 72, z = 49
   20. many possible answers, for example a road passing under a railroad track
   21. many possible answers, for example the north and south bound lanes of an
       interstate highway
   22. the lines never meet; they lie in different planes

Unit 2, Lesson 4

   1.   parallel
   2.   neither
   3.   perpendicular
   4.   perpendicular
   5.   parallel
   6.   neither
   7.   perpendicular

Unit 3 Lesson Answers

Unit 3, Lesson 3

   1. 94 ½o
   2. 26, 78, 36, 66
   3. 48
   4. 44, 136
   5. 65, 27
   6. 44
   7. 23 4/7, 132 6/7
   8. 180, the sum of the measures of the angles of a triangle is 180
   9. 180, linear pairs are supplementary
   10. 360, the sum of the measures of the exterior angles of a polygon is 360

Unit 3, Lesson 4
   1.   64, 52
   2.   58, 58
   3.   30,75,75
   4.   11

Unit 3, Lesson 5

   1.   no, 2 + 3 is not greater than 5
   2.   no, 1 + 2 is not greater than 4
   3.   no, 6 + 1 is not greater than 8
   4.   yes, 5 + 5 is greater than 5
   5.   yes, 2x + 3x (5x) is greater than 4x

The Triangle Inequality Theorem states that the sum of the measures of any two sides of
a triangle must always be greater than the measure of the third side.
Unit 3 Lesson 6

   1. 20, 45, 45
      Solution: 5x – 10 = 90o
                    + 10 + 10
                       5x = 100
                        5      5
                          x = 20
                  ABD = 2(20) + 5 = 45
                 BAD +  ADB +  ABD = 180
                 BAD + 90 + 45 = 180
                 BAD = 45
   2. 17, 46, 92 (Use a similar method to one)
   3. 2, 12, 24 (Use a similar method to one)
   4. 2.5, 4.5, 9, 90o (Use a similar method to one)
   5. In a right triangle two of the altitudes are the legs of the triangle, the third altitude
      and all of the other special segments are in the interior of the triangle. In an obtuse
      triangle two of the altitudes lie outside the triangle, the third altitude and all of the
      other special segments are in the interior of the triangle. In an acute triangle all of
      the special segments lie inside the triangle.


Unit 4 Lesson Answers

Unit 4 Lesson 2

   1. 15
   2. 13
   3.   5
   4. 3 10 or      90
Unit 4 Lesson 3

   1.   obtuse, 100 > 25 + 49
   2.   obtuse, 144 > 64 + 64
   3.   acute, 16 < 9 + 9
   4.   right, 100 = 36 + 64
   5.   acute, x2 < x2 + x2

Unit 4, Lesson 4

   1.   8, 8 2
   2.   6.3, 6.3 2
   3.   17/ 2, or (17 2)/2
   4.   2, 2
   5.   3 3, 6
   6.   14.5, 14.5 3
   7.   5.3/ 3 or (5.3 3)/3, 10.6/ 3 or (10.6 3)/3
   8.   3.95, 3.95 3

Unit 4 Lesson 5

   1. the length of the opposite side/the length of the hypotenuse
   2. the length of the adjacent side/the length of the hypotenuse
   3. the length of the opposite side/the length of the adjacent side
   4. tan 42 = x/16.4
   5. cos 39 = x/12
   6. sin 64 = x /19.3
   7. tan x = 26/19
   8. tan 63 = a/28, a = 55
   9. cos 29 = 13.2/c, c = 15
   10. 46o

Unit 5 Lesson Answers

Unit 5, Lesson 2

   1.   6, 90o, 4, 20 or 2 5, 4 5
   2.   90o, 5 2, 2.5 2, 5 2, 45o
   3.   1in., 6in., 62o
   4.   90o, 4 ft., 7 ft., 65, 65

Unit 5, Lesson 4

   1. V = 297.5 m3
   2.   V = 272 in3 or about 854.5 in3
   3.   V = 75 in3 or about 235.6 in3
   4.   V = 93.3 ft3
   5.   V = 288 cm3 or 904.8 cm3

Unit 6 Lesson Answers

   1.   c = 2, d = 7.5,  F = 36o
   2.   yes by SAS
   3.   no
   4.   no
   5.   yes by AA
   6.   yes by SSS

Unit 6, Lesson 3

   1. r = 3 cm
   2. x = 100 cm3
   3. a = 2.5 cm 2