# Properties of Triangles by Y77X425

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```									Geometry                                Mrs. Franks                               10th Grade

Unit Plan Information

Properties of Triangles

Jessica Franks

Mathematics
Geometry

Learning Objectives: See daily lessons

Essential Questions: What are the properties of a triangle?
How can I identify congruent triangles?
What is an altitude?
What are the different types of triangles?
What are corresponding parts?
What happens when the medians of a triangle meet?
What happens when the altitudes of a triangle meet?
When you connect the midpoints of a triangle what do you get?

Enduring Understandings:
Students are able to geometric relationships are evident in real-life
situations.
Students will be able to recognize math processes in the future and be able
to locate appropriate resource materials to assist them.
Students will be able to recognize reasoning and proof as fundamental
aspects of mathematics.
Students will be able to see relationships that exist between the angles and
sides of geometric figures can be proven.

At the conclusion of this unit the students should be able to use properties, theorems and
postulates to prove the congruency of triangles to one another.

Instructional Procedures: See Daily Lesson Plans

Standards:
NY State                                        NY State
Geometry Standards                             Technology Standards

   G.G.28 Determine the congruence of               1.a Students demonstrate a sound
two triangles by using one of the five            understanding of the nature and
congruence techniques (SSS, SAS,                  operation of technology systems
ASA, AAS, HL), given sufficient                  2.b Students practice responsible use of
information about the sides and/or                technology systems, information, and
angles of two congruent triangles                 software.
   G.G.29 Identify corresponding parts of           2.c Students develop positive attitudes
congruent triangles                               toward technology uses that support
   G.G.30 Investigate, justify, and apply            lifelong learning, collaboration,
theorems about the sum of the                     personal pursuits, and productivity.
measures of the angles of a triangle             3.a Students use technology tools to
   G.G.31 Investigate, justify, and apply            enhance learning, increase productivity,
the isosceles triangle theorem and its            and promote creativity.
converse                                         5.b Students use technology tools to
   G.G.32 Investigate, justify, and apply            process data and report results.
theorems about geometric inequalities,           6.a Students use technology resources
using the exterior angle theorem                  for solving problems and making
   G.G.33 Investigate, justify, and apply            informed decisions.
the triangle inequality theorem                  6.b Students employ technology in the
   G.G.34 Determine either the longest               development of strategies for solving
side of a triangle given the three angle          problems in the real world.
measures or the largest angle given the
lengths of three sides of a triangle
   G.G.43 Investigate, justify, and apply
theorems about the centroid of a
triangle, dividing each median into
segments whose lengths are in the ratio
2:1
   G.G.44 Establish similarity of
triangles, using the following theorems:
ASA, SAS, and SSS
   G.G.45 Investigate, justify, and apply

DATE       OBJECTIVE:                                  DATE    OBJECTIVE:

12/05      - Understand the key properties of          12/05   - Identify congruent figures and
triangles using geometer’s                          corresponding parts.
sketchpad (GS)                                      - Prove that two triangles are
- Classify triangles by their sides                 congruent.
and angles                                          - Prove that triangles are congruent
using the SSS and SAS Congruence
Postulates.

CONTENT:                                            CONTENT:

- Triangles                                         - 4.2: Congruence and Triangles
- 4.1: Triangles and Angles                         - 4.3: Proving Triangles are
Congruent: SSS and SAS

ACTIVITIES:                                         ACTIVITIES:

- Class discussion on Triangles                     - Go to computer lab
- What make a triangle a triangle?                  - Bell Ringer
- Go to the computer lab                            - Go over homework
- Introduce geometer’s sketchpad                    - Using GS have class take notes,
to the students                                     practice, explore and discuss
- Allow students to get the used to                 congruency of triangles
the new program by letting them
explore
explore basic properties of triangles

MATERIALS NEEDED:                                   MATERIALS NEEDED:

Calculators, Worksheet for GS from                  Calculator, Compass
http://sierra.nmsu.edu/morandi/Cour
(In Notes)

ASSESSMENT:                                         ASSESSMENT:

Student responses – verbal and                      Student responses – verbal and
written. Class participation –                      written. Class participation –
Sketch. Homework assignment.                        Sketch. Homework assignment.

PRACTICE:                                           PRACTICE:

In class - sketches                                 In class - see written examples from
Homework – Triangle Worksheet                       notes
Homework – Explore applet at
http://illuminations.nctm.org/tools/t
ool_detail.aspx?id=4 .

DATE       OBJECTIVE:                                  DATE    OBJECTIVE:

12/05      - Prove that triangles are congruent        12/05   - Identify the mid-segments of a
using the ASA Congruence                            triangle.
Postulate and the AAS Congruence                    - Use properties of mid-segments of a
Theorem.                                            triangle.
- Use properties of medians of a                    - Use triangle measurement to decide
which side is longest or which angle is
triangle.
largest.
- Use properties of altitudes of a                  - Use the triangle Inequality.
triangle.

CONTENT:                                            CONTENT:

- 4.4: Proving Triangles are                        - 5.4: Mid-segment Theorem
Congruent: ASA and AAS                              - 5.5: Inequalities in One Triangle
- 5.3: Medians and Altitudes of a
Triangle

ACTIVITIES:                                         ACTIVITIES:

- Go to computer lab                                - Go to computer lab
- Bell Ringer                                       - Bell Ringer
- Go over homework                                  - Go over homework
the class take notes, practice,                     have the class take notes, practice
explore and discuss congruency,                     and discuss Mid-segment and
medians and altitudes of triangles                  inequalities in one triangle.

MATERIALS NEEDED:                                   MATERIALS NEEDED:

Calculator, Worksheet for GS from                   Calculator, Sketch from Key
http://sierra.nmsu.edu/morandi/Cour                 Curriculum Press on web page
Notes)

ASSESSMENT:                                         ASSESSMENT:

Student responses – verbal and                      Student responses – verbal and
written. Class participation –                      written. Class participation –
Sketch. Homework assignment.                        Sketch. Homework assignment.

PRACTICE:                                           PRACTICE:

In class - sketches                                 In class - see written examples from
Homework – Textbook Problems                        notes
Homework - Textbook problems.

DATE       OBJECTIVE:                                  DATE    OBJECTIVE:

12/05      - Write the equation of a line given        12/05   - Assess Knowledge of Students
a point on the line and the slope of
the line
- Write the equation of a line given
two points on the line

CONTENT:                                            CONTENT:

- 5.3 & 5.5: Writing equations of                   - Chapters 4 & 5
lines with two points and Point-
Slope Form

ACTIVITIES:                                         ACTIVITIES:

- Bell Ringer                                       - Go to computer lab
- Go over homework                                  - Bell Ringer
- Using the graphing calculator                     - Go over homework
have class take notes, practice and                 - Using Geometer’s sketchpad
discuss the equations of lines.                     assess triangles
- Complete Worksheet on the
writing equations of lines

MATERIALS NEEDED:                                   MATERIALS NEEDED:

Computers, Calculators, notes                       Computers, Geometer’s Sketchpad,
Calculators, Teacher created exam

ASSESSMENT:                                         ASSESSMENT:

Student responses – verbal and                      Student responses – verbal and
written. Class participation –                      written. Class participation –
Sketch. Homework assignment.                        Sketch. Teacher created exam

PRACTICE:                                           PRACTICE:

In class - see written examples from                In class - Teacher created exam
notes                                               Individually - Teacher created
Homework – Textbook problems                        exam

Chapters
4&5
Course 2R
Mrs. Franks

Triangles
What do you remember about triangles?

In this assignment we will learn how to use the program Geometer's Sketchpad. This program is
very useful for learning about geometry. We will discover several geometric facts this semester
through its use.

Here are several tasks to perform in Geometer's Sketchpad. You should use the
program enough to be able to do these tasks with ease. When you open the
program, you will see six icons on the left side of the screen. They are, from top to
bottom, the arrow tool, the point tool, the compass (or circle) tool, the
straightedge tool, the text tool, and the custom tool. The arrow tool is used to
select objects. The next three are used to draw points, circles, and lines.

One important thing to know about is how to highlight objects. By clicking on an
object it will be highlighted, and then can be used in further constructions. The
order in which you highlight objects can affect the resulting construction.

   Draw a point: Click on the point tool, then click where you want a point.
   Draw a line segment: Click on the line tool. The icon should show two points and a segment
connecting them. To draw a line segment click the mouse where you want the segment to
begin, and holding the mouse, drag it until you get to where you want the line to end, then
release the mouse.
   Draw a ray and line: Click and hold the mouse on the line tool until you see three icons.
These, from left to right, are the line segment, ray, and line tools. Click on the appropriate
one, then click and hold the mouse somewhere on the screen, then drag to get the ray or
line.
   Draw a circle: Click on the circle tool, then click and hold the mouse, move to size the
circle. Alternatively, if you want a circle centered at a given point, with the circle tool, place
the cursor over the point and then draw the circle. If you want the circle centered at a
certain point and passing through another point, click on the center and then click on the
second point. Finally, click on construct, then circle by center and point. See what happens if
you highlight the points in reverse order and construct the circle by center and point.
Circles are determined by two points, one being the center and the other being a point on
the circle.
   Resize the circle: Click on the arrow tool, then on the point on the circle. Drag this point to
resize the circle. Alternatively, click and drag the center.
   Move the circle: Click on the arrow tool, then on the circle away from the point on the
circle. Drag to move the circle.
   Draw a triangle: using the line segment tool, draw a line segment. Then draw a second
segment starting where the first segment ended. Finally, draw a third segment starting
where the second segment ended and ending where the first segment started.
   Resize the triangle: Click the mouse on the arrow tool. Then click on one of the vertices of
the triangle (i.e., one of the endpoints), then drag the mouse to resize. Alternatively, click
and drag one of the sides.

   Move the triangle: Click the arrow tool. Then click on two of the sides (or the three
vertices). Then drag one of the sides.
   Measure the angles of the triangle: Click the arrow tool. Then click three of the vertices
in order. Then go to Measure, Angle.
   Select more than one object: Click on the arrow tool. Click on the objects you wish to
select. You should see which objects are selected.
   Draw the interior of a triangle: Click on the arrow tool. Then click on all three vertices of
the triangle. You should see large dots over each of them. Click the mouse on the menu item
construct, then on polygon interior.
   Draw a four-sided figure: Once you have drawn it, resize it by moving one of the vertices.
Notice that you can make many different shapes.
   Draw the four-sided figure's interior.
   Draw an angle bisector. Geometer's Sketchpad views an angle as three points selected in
order. The middle point is the vertex, or corner, of the angle. You can then draw the angle
by drawing rays from the vertex through the other two points. Once you have drawn and
selected three points, click on construct, and then angle bisector. This line should cut the
angle into two equal pieces. If it does not appear to do so, look carefully at the order in
which you selected your three points, since there are three different angles that can be
made from the three points (the three angles of the triangle formed by the three points).
   Find the intersection of two lines, segments, or circles: Draw two line segments (or rays
or lines or circles) that cross. With the point tool, put the mouse over the intersection and
click. Move one of the line segments and watch what happens to the intersection point.
Alternatively, select both line segments, then click on construct, then on intersection.
   Draw perpendicular and parallel lines: Draw a line. Select the line and a point on the line.
Then click construct, then perpendicular line. This constructs a line perpendicular to the
given line and passing through the given point. Next, plot a point off of a given line. Select
the line and the point. Click construct, then parallel line. This produces a line through the
given point and parallel to the given line.
   Label points or sides: Click on the label tool (the one that looks like a hand), then click on
whatever you want to label. If you want to change the label, double click on the label (after
selecting either the label tool or the arrow tool).
   Open documents: Open the file Square.gsp. It is on my web page
instructions once you open it and play around with them accordingly.
   Print documents: Click on file, then on print preview. Click on fit to page if it shows your
sketch printing on two pages . Finally, click print. If you click print directly, your document
may print on two pages.

4.2 Congruence and Triangles

   Two Geometric Figures are __________________________ if they have exactly the

same _____________________ and ___________________________.

   When two figures are ________________________, there is a correspondence between

their angles and sides such that, corresponding ____________________ are congruent

and corresponding ________________________ are congruent.

For the triangles below you can write ABC  PQR
A
Corresponding Angles               Corresponding Sides

B                              C

P

Q                              R

Using Geometer’s Sketchpad: Create Two Congruent Triangles. Show that Corresponding
Angles are Congruent and Corresponding Sides are Congruent (Using the Measure Tool).

Example 1: Congruent Figures

In the diagram NPLM  EFGH                        P
E                  F
a. Find the value of x              8m                        L       (7y + 9)º
72º
b. Find the value of y
N   110º
10m        H                                  G
(2x – 3) m
87º

M

Theorem 4.3 Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles
are also congruent.

If A  D and B  E then C  F                                        E

B

D
F
A                                          C

Use Geometer’s Sketchpad: Create two triangles (NOT Congruent). Measure all of the angles
in each triangle. Now move your points around so that you have two sets of angles congruent.
Is the third set of angles congruent?

Example 2: Find the value of x if MNL  TRS :
M
T       (2x + 30)º

N
55º
R

65º

L                              S

4.3 Proving Triangles are Congruent: SSS and SAS

Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of
the sides of the two triangles. Now move the triangles such that the sides of the first triangle are
congruent to the sides of the second triangle. Now without moving the triangles measure all the
angles of both triangles. What do you notice?

Postulate 19 Side – Side – Side (SSS) Congruence Postulate

    If three sides of one triangle are congruent to three sides of a second triangle, then the

two triangles are ___________________________.
N
If          Side     MN  QR
Side NP  RS
Side PM  SQ
Then        MNP  QRS                   R
M                                        P

Q
S

Using a Compass Construct a triangle that is congruent to the given triangle ABC.

A

C                                 B

Now that we’ve used the compass try using Geometer’s Sketchpad to construct congruent
triangles. Remember, you must show your arcs to have a valid construction. Hint use construct
a circle.

Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of
two sides and the angle between the two sides of the two triangles. Now move the triangles such
that these three measurements are congruent to each other. Now without moving the triangles
measure the rest of the sides and angles of both triangles. What do you notice?

Postulate 20 Side – Angle – Side (SAS) Congruence Postulate

   If two sides and the included angle of one triangle are congruent to two sides and the

included angle of a second triangle, then the two triangles are __________________.

X
If       Side PQ  WX                                                                 Q
Angle Q  X
Side     QS  XY
P                                          W                Y
Then              PQS  WXY                                                                      S

Example 3: Use the SSS Congruence Postulate to Prove the two triangles congruent.

8

6

B: (-7.00, 5.00)        A: (-4.00, 5.00)                              F: (6.00, 5.00)
B                   A                                           F

4

2                               E: (6.00, 2.00)
D                   E

D: (1.00, 2.00)

-10           C           -5                                             5                          10

C: (-7.01, 0.00)

-2

-4

-6

-8

Homework: Go to http://illuminations.nctm.org/tools/tool_detail.aspx?id=4 and play around
with the applet. Answer the questions at the bottom of the page and print out your explorations.

4.4 Proving Triangles are Congruent: ASA and AAS

Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of
two angles and the side between the two them in both triangles. Now move the triangles such
that these three measurements are congruent to each other. Now without moving the triangles
measure the rest of the sides and angles of both triangles. What do you notice?

Postulate 21 Angle – Side – Angle (ASA) Congruence Postulate

If two ______________ and the included _______________ of one triangle are congruent to two
angles and the included side of a second triangle, then the two triangles are _________________.

If     Angle         A  D                      B                                              E
Side           AC  DF
Angle         C  F
Then               ABC  DEF                    C                                                  F
A       D

Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of
two angles and a side NOT between the two angles in both triangles. Now move the triangles
such that these three measurements are congruent to each other. Now without moving the
triangles measure the rest of the sides and angles of both triangles. What do you notice?

Theorem 4.5 Angle – Angle – Side (AAS) Congruence Theorem

If two _________________ and a non-included ______________________ of one triangle are
congruent to two angles and the corresponding non-included side of a second triangle, then the
two triangles are ________________________.
A                 F
If     Angle         A  D                                                                         E
Angle         C  F
Side           BC  EF
Then               ABC  DEF

C                     B           D

This is Wonderful that Geometer’s Sketchpad is working to show us these postulates and
theorems are true, but can anyone tell us why, or show us another way using Geometer’s
Sketchpad to prove these postulates to us?

- With a partner try to find another way to use geometer’s sketchpad to prove these to the class.

Example 1: Is it possible to prove that the triangles are congruent? If so, state the postulate or
theorem you would use. Explain your reasoning.

M                                N
D                      A               F                  G
2
3

C

1
4

E                                                                    P                                O
B          I                   H

D  B                                                        MN PO and PM ON

Example 2: You want to describe the boundary lines of a triangular piece of property to a
friend. You fax the note and the sketch below to your friend. Have you provided enough
information to determine the boundary lines of the property? Use Geometer’s Sketchpad to
explain.

N
The southern border is a line running                    cherry tree
east from the apple tree, and the
western border is the north – south
line running from the cherry tree to                                           250ft
the apple tree. The bearing from the
easternmost point to the northernmost
point is W 53.1º N. The distance
between these points is 250 ft.                                                         53.1º

5.3 Medians and Altitudes of a Triangle

Median of a Triangle – a segment whose endpoints are a ____________ of the triangle

and the ____________________ of the opposite side.

A

C                M             B

Use Geometer’s Sketchpad: Construct a Triangle. Find the midpoint of Each Side. Now
connect the vertex of each angle to the midpoint on the opposite side. What do you notice?

Drag one vertex of the triangle to see an acute, obtuse and right triangle. What do you notice
now?

The medians of a triangle are __________________________.

Concurrent Lines – Lines that intersect at ____________________________________.

The point of concurrency is called the _________________________ of the triangle.

Use Geometer’s Sketchpad: Construct a point at the centroid. Now use the Measure Tool to
measure the distance from each vertex to the centroid. Use the Calculate Tool to find the ratio of
each Median. What do you notice?

Theorem

Theorem 5.7 Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point that is two thirds of the distance from each
B
vertex to the midpoint of the opposite side.
D
If P is the centroid of ΔABC, then                           P
E
2           2             2
AP  AD, BP  BF, andCP  CE.                                                       C
3           3             3
F

A

Example 1: P is the centroid of ΔQRS shown below and PT = 5, find RT and RP.

Q

P

S                                            R

Example 2: Find the coordinates of the centroid of ΔJKL.

12

L
10

8

6
K

4

2
J

5                10

Altitude of a Triangle – the ______________________ segment from a vertex to the opposite
side or to the line that contains the opposite side.

Use Geometer’s Sketchpad: Construct a triangle. Construct perpendicular segments from a
vertex to the opposite side of the triangle. Repeat for all three sides. Do these lines intersect? If
they do construct a point at the intersection. Drag one of the vertices of the triangle, What do
you notice about the point of intersection? Think about the following questions.

The altitude of a triangle can be where?

How many altitudes does a triangle have?

Are the lines concurrent?

The point where they intersect is called the _______________________________________.

Example 3: Where is the orthocenter located in each type of triangle? Use Geometer’s
Sketchpad to see the sketch. Try to draw it.

a. Acute Triangle              b. Right Triangle              c. Obtuse Triangle

Theorem

Theorem 5.6 Concurrency of Altitudes of a Triangle                           F
A                     B
The lines containing the altitudes of a triangle are concurrent.
H
If AE , BF and CD are the altitudes of                               D            E

ΔABC, then the lines AE , BF and CD
intersect at some point H.                                                    C

5.4 Mid-segment Theorem

A Mid-segment of a triangle is a segment that __________________________________ of two
sides of a triangle.

Example 4: Using Geometer’s Sketchpad Show that the mid-segment MN is parallel to side
JK and is half as long. Hint: How do we know lines are parallel?

8

6

K
4

J
2

-5                                    5

L
-2

Draw in the missing pieces (segments and measurements) from Sketchpad.

Midsegment Theorem
C
Theorem 5.9 Mid-segment Theorem

The segment connecting the midpoints of
two sides of a triangle is parallel to the                D           E
third side and is half as long.

1
DE AB and DE               AB                     A
2                                            B

Example 5:
UV and VW are mid-segments of ΔRST. Find UW and RT. If RS  12 and VW  8

R

U

V
T

W

S

5.5 Inequalities in One Triangle

Theorems
B
Theorem 5.10
If one side of a triangle is longer than                                                        5
another side, then the angle opposite                                  3
the longer side is larger than the angle
opposite the shorter side.                                         A                                       C

Theorem 5.11
D                                                E
If one angle of a triangle is larger than                              60º                          40º
another angle, then the side opposite
the larger angle is longer than the side
opposite the smaller angle.

F

Largest Angle

Shortest
Side

Longest Side                                                                         Smallest Angle

Example 1: Write the measurements of the triangles in order from least to greatest.

a.                                                    R
b.                  F

100
8
7
35
45                                         H
5               Q
P                                                                          G

Use Geometer’s Sketchpad: Construct a ray. Construct a point above the ray and a point on
the ray. Construct a triangle using the endpoint of the ray and the two new points that you have
created. Measure the exterior and interior angles. Play around with the calculations. Do you
notice anything?

B

A                  C        D

Theorem
A
Theorem 5.12 Exterior Angle Inequality
The measure of an exterior angle of a
triangle is greater than the measure of
either of the two nonadjacent interior
angles.
m1 > mA and m1 > mB                                     1
D           C                          B

Use Geometer’s Sketchpad: Go to my webpage
http://www.bataviacsd.org/webpages/JFranks/course__3r.cfm?subpage=6660 and open
Inequalities in One Triangle. Keep clicking random break and see if you can make a triangle.
What do you notice about the lengths of the sides when you can and cannot make a triangle?

Theorem

Theorem 5.13 Triangle Inequality
The sum of the lengths of any two sides of a triangle
is greater than the length of the third side.

AB + BC > AC
AC + BC > AB
AB + AC > BC

Example 3: A triangle has one side of 10 centimeters and another of 14 centimeters.
Describe the possible lengths of the third side.

5.3 & 5.5 Writing equations of lines with two points and Point-Slope Form

What two pieces of information do you need to write the equation of a line?

The __________________________ and the _______________________________.

What is the Slope – Intercept Form of a line? ________________________________.

What is the Slope Formula? __________________________________.

Example 1: Write the equation of the line that passes through the points (1, 6) and (3, -4).

What do we need to write the equation? _____________________ and ____________________

What can we find with two points? ____________________________________

Writing an equation of a line given two points

Step 1 Find the Slope. Substitute the coordinates of the two given points into the formula for
y  y1
slope, m  2         .
x 2  x1

Step 2 Find the y-intercept. Substitute the slope m and the coordinates of one of the points
into the slope-intercept form, y = mx + b.

Step 3 Write an equation of the line. Substitute the slope m and the y-intercept b into the
slope-intercept form, y = mx + b.

Another strategy for writing the equation of a line is __________________________________.

Point – Slope Form of the equation of a line

The point – slope form of the equation of the nonvertical line that passes through
a given point ( x1 , y1 ) with a slope of m is

y  y1  m(x  x 1 )

2
Example 2: Write an equation of the line given the point (2, 5) and a slope m of     .
3

Example 3: Write an equation of a line given the points (-2, 3) and (-1, 1).

Another way to write the equation of line when given two points is to use your graphing
Calculator. Let’s use the last example:
Write an equation of a line given the points (-2, 3) and (-1, 1).

1. First hit the STAT button and then Edit.             5. Then GRAPH it.

2. Enter your x-values into L1 and your                 6. Now we want the equation of the line.
y-values into L2                                        Go back to STAT, CALC

7. We want the equation of a line #4. Then
nd
3. Now hit 2 , Y = to get into STAT PLOT                type L1, L2 , Y1 (Under VARS).
and then ENTER

4. Turn on your STAT PLOT                               8. Hit ENTER and your coefficients will
appear and you can look at the graph to see