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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

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