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Unit 4 Congruency and Triangle Proofs

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Triangle Congruence and Rigid Transformations In the diagram at the right, a transformation has occurred on ABC. Describe a transformation that created image ABC from ABC. Is ABC congruent to ABC? Explain. The vertices of MAP are M(-8, 4), A(-6, 8) and P(-2, 7). The vertices of MAP are M(8, -4), A(6, -8) and P(2, -7). Plot MAP. Verify that the sides of the triangles are congruent. Describe a rigid motion that can be used to MAP Given PQR with P(-4, 2), Q(2, 6) and R(0, 0) is congruent to STR with S(2, -4), T(6, 2) and R(0, 0). Plot STR. Describe a rigid motion which can be used to verify the triangles are congruent.

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Given RST with R(1, 1), S(4, 5) and T(7, 5). Plot the reflection of RST in the y-axis and label it RST. Is RST congruent to RST? Explain. Plot the image of RSTunder the translation (x, y) (x + 4, y – 8). Label the image of RST. Is RST congruent to RST? Explain. Is RST congruent to RST? Explain. Given DFE with D(1, -1), F(9, 6) and E(5,7) and BAT with B(1, 1), A(-6, 9) and T(-7, 5). Describe a transformation that will yield BAT as the image of DFE. Is BAT congruent to DFE? Explain. Given CAP with C(-4, -2), A(2, 4) and P(4, 0) and SUN with S(-8, -4), U(4, 8) and N(8, 0). Describe a transformation that will yield SUN as the image of CAP. c) Is CAP congruent to SUN? Explain.

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Discovering Congruent Triangles Activity

Part 1 1. Have students put the 3 straws of different lengths together

to form a triangle as shown. 2. Form another triangle with the other set of straws. 3. Measure the angles of both triangles using a protractor. Questions:

1. What are the measures of the 3 angles in the first triangle? 2. What are the measures of the 3 angles in the second triangle? 3. What is the relationship between the angles of each triangle? 4. Are the triangles congruent? 5. Can the straws be rearranged to form a triangle with different angles?

Part 2 1. Take 2 of the straws, place them on a piece of paper, and form a 60

degree angle between them. 2. Take the 2 straws of the same length and also form a 60 degree angle

between them. 3. Draw a line to represent the 3rd side. Repeat the process for the 2nd

triangle. 4. Measure the length of the 3rd side and the two remaining angles for each

triangle. Questions:

1. What is the length of the 3rd side? 2. What are the measures of the remaining angles? 3. Are the two triangles congruent? 4. Use any two straws and any angle of your choice. Do you get the same result? Will you always get the same result?

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Part 3 1. Measure three angles measuring 80, 60, and 40 degrees on the corners

of 2 pieces of construction paper or cardstock, cut them out, and label them.

2. On a piece of paper, take one of the straws, and place two of the cut-out angles on each end as shown. Repeat the process for the 2nd triangle.

3. Using a ruler, draw a segment along each of the angle. The two segments should intersect forming the last angle. Repeat the process for the 2nd triangle.

4. Measure the 3rd angle and the lengths of the 2 sides in each triangle. Questions:

1. What is the measure of the 3rd angle for each triangle? 2. What are the measures of the remaining 2 sides for each triangle? 3. Are the triangles congruent? 4. What if you used the 5cm straw? The 8cm straw? A straw with a different length?

Part 4

1. Use two of the angles used in the example above. 2. Use one of the straws and place one of the angles alongside it as

shown. Draw a long segment like the dashed one in the drawing. Repeat the process for the 2nd triangle.

3. Place the second angle along this segment so that when a 2nd segment is drawn, it will connect with the end of the straw.

4. Measure the 3rd angle and the two remaining sides. Questions:

1. What is the measure of the 3rd angle for each triangle? 2. What are the measures of the remaining 2 sides for each triangle? 3. Are the triangles congruent?

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Part 5

1. Place two of the straws together forming an angle of any degree for one triangle, and repeat the process for the 2nd triangle.

2. Use one of the pre-cut angles and place alongside the longer of the sides but not as the included angle.

3. Draw a segment to connect the 3rd side to the other two sides.

4. Swing the 8cm straw so that it hits the 3rd side at a different spot in the 2nd triangle as in the first.

5. Measure the 3rd side and the remaining 2 angles in each triangle.

Questions

1. What is the measure of the 3rd side for each triangle? 2. What are the measures of the remaining 2 angles for each triangle? 3. Are the two triangles congruent? 4. Do you think that you would get different results if you used a different angle?

Part 6

1. Place the 3 angles so that they can form a triangle without measuring the sides initially. Draw segments connecting the angles. Repeat the process for the second triangle.

2. Measure the 3 sides for each triangle. Questions

1. What are the measures of the 3 sides for each triangle? 2. Are the two triangles congruent?

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Congruent Triangles Investigation Part I What does it mean to say two triangles are congruent? List the ways to justify that triangles are similar. Examine the triangles with all side lengths labeled. Are they similar? Why? What is the scale factor? _______ : _______ What do we know about the corresponding angles of similar triangles? What does this tell us about the pair of triangles? Part 2: Examine the triangles with two sides lengths and an included angle labeled. Are they similar? Why? What is the scale factor? _______ : _______ Since the triangles are similar, what do we know about P and H? What do we know about I and O? Use the scale factor you gave in part b to determine the length of 𝑂𝐻̅̅ ̅̅ ̅. What does this tell us about the pair of triangles? Part 3: Examine the triangles with two angle pairs marked congruent. Are they similar? Why? What is the scale factor? _______ : _______ Since the triangles are similar, what do we now bout K and Y? Use the scale factor you gave in part c to determine the lengths of 𝐾𝐿̅̅ ̅̅ and 𝑌𝑍̅̅ ̅̅ . What does this tell us about the pair of triangles?

4 cm

2.15 cm

4 cm

3.2 cm

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Think back to the three situations we examined. In #1, we were given 3 pairs of sides of one triangle are congruent to 3 pairs of sides of another triangle. The triangles are congruent by the

___________, __________, __________ (SSS) Postulate. In #2, we were given 2 pairs of sides and an included angle of one triangle are congruent to 2 pairs of sides and an included angle of another triangle. The triangles are congruent by the

___________, __________, __________ (SAS) Postulate. In #3, we were given 2 pairs of angles and an included side of one triangle are congruent to 2 pairs of angles and an included side of another triangle. The triangles are congruent by the

___________, __________, __________ (ASA) Postulate. Two other Postulates

Angle, Angle, Side Postulate (AAS Theorem) How can you change this into one of the Postulates that we already have?

Hypotenuse, Leg Theorem (HL Theorem) By Pythagorean Theorem, if you know the hypotenuse and 1 leg, you can calculate the 2nd leg by _________________ Theorem and prove congruence by ____________ postulate. Note: Postulate – Statement which is taken to be true without proof. Theorem – Statement that can be demonstrated to be true by accepted mathematical operations and arguments.

3 cm2 cm

4 cm4 cm

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HW: How do you prove triangles are congruent? #1 & 2 Use the given coordinates to determine if ABC DEF

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CPCTC Essential question: What can you conclude about two triangles that are congruent? When you know that two triangles are congruent, you can make conclusions about the sides and angles of the triangles.

Reflect: If you know that ABC DEF, what six congruent statement about segments and angles can you write? Why?

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When two triangles are congruent the corresponding parts are the sides and angles that are images of each other. You write congruence statements for two figures by matching the corresponding parts. In other words, the statement ABC DEF contains the information that 𝐴𝐵̅̅ ̅̅ corresponds to 𝐷𝐸̅̅ ̅̅ so that 𝐴𝐵̅̅ ̅̅ ≅ 𝐷𝐸̅̅ ̅̅ , A corresponds to D so that A D, and so on.

Corresponding Parts of Congruent Triangles are Congruent Theorem (CPCTC)

If two triangles are congruent, then the corresponding sides are congruent and the corresponding angles are congruent.

Converse of the Corresponding Parts of Congruent Triangles are Congruent Theorem

(CPCTC)

If tow triangles have corresponding sides that are congruent and the corresponding angles that are congruent, then the triangles are congruent.

Examples

Discuss:

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CPCTC and Naming Congruent Triangles I. Draw and label a diagram. Then solve for the variable and the missing measure or length. 1. If ∆𝐵𝐴𝑇 ≅ ∆𝐷𝑂𝐺, and 𝑚∠𝐵 = 14, 𝑚∠𝐺 = 29, 𝑎𝑛𝑑 𝑚∠𝑂 = 10𝑥 + 7. Find the value of x 𝑚∠𝑂.

x = ___________ 𝑚∠𝑂= _________ 2. If ∆𝐶𝑂𝑊 ≅ ∆𝑃𝐼𝐺, and 𝐶𝑂 = 25, 𝐶𝑊 = 18, 𝐼𝐺 = 23, 𝑎𝑛𝑑 𝑃𝐺 = 7𝑥 − 17 . Find the value of x and PG.

x = ___________

PG=___________ 3. If ∆𝐷𝐸𝐹 ≅ ∆𝑃𝑄𝑅 and 𝐷𝐸 = 3𝑥 − 10, 𝑄𝑅 = 4𝑥 − 23, 𝑎𝑛𝑑 𝑃𝑄 = 2𝑥 + 7. Find the value of x and EF.

x = ___________ EF = __________ II. Use the given information and triangle congruence statement to complete the following. 4. ∆𝐴𝐵𝐶 ≅ ∆𝐺𝐸𝑂, AB = 4, BC = 6, and AC = 8. What is the length of 𝐺𝑂̅̅ ̅̅ ? How do you know?

5. ∆𝐵𝐴𝐷 ≅ ∆𝐿𝑈𝐾, 𝑚∠𝐷 = 52°, 𝑚∠𝐵 = 48°, 𝑎𝑛𝑑 𝑚∠𝐴 = 80°.

a. What is the largest angle of ∆𝐿𝑈𝐾?

b. What is the smallest angle of ∆𝐿𝑈𝐾?

6. ∆𝑆𝑈𝑁 ≅ ∆𝐻𝑂𝑇. ∆𝑆𝑈𝑁 is isosceles. Is there enough information to determine if ∆𝐻𝑂𝑇 is isosceles?

Explain why or why not.

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III. Complete the congruence statement for each pair of congruent triangles. Then state the reason you are able to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write “none” in the blanks.

7. ∆𝐸𝐹𝐷 ≅ ∆___________ 8. ∆𝐴𝐵𝐶 ≅ ∆___________ 9. ∆𝐿𝐾𝑀 ≅ ∆___________ by ________ by ________ by ________

10. ∆𝐴𝐵𝐶 ≅ ∆___________ 11. ∆𝐴𝐵𝐶 ≅ ∆___________

by ________ by ________ IV. Use the given information to mark the diagram and any additional congruence you can determine from the diagram. Then complete the triangle congruence statement and give the reason for triangle congruence . 12. 13. Given: ∠1 ≅ ∠3, ∠2 ≅ ∠4 Given: ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷, ∠𝐴𝐷𝐵 ≅ ∠𝐶𝐷𝐵 ∆𝐴𝐵𝐶 ≅ ∆__________ by __________ ∆𝐴𝐵𝐷 ≅ ∆__________ by __________ 14. 15. Given: 𝐺 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐹𝐵̅̅ ̅̅ 𝑎𝑛𝑑 𝐸𝐴̅̅ ̅̅ Given: ∠1 ≅ ∠3, 𝐶𝐷̅̅ ̅̅ ≅ 𝐴𝐵̅̅ ̅̅ ∆𝐴𝐵𝐺 ≅ ∆__________ by __________ ∆𝐴𝐵𝐶 ≅ ∆__________ by __________

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Congruent Triangle Problems - Honors I. Δ𝑃𝑄𝑅 ≅ Δ𝐴𝐵𝐶. Find the values of x and y.

1. 𝑚∠𝑅 = 5𝑥 + 70, 𝑚∠𝐶 = 24𝑥 − 25, 𝑄𝑅 = 4𝑦 + 2, 𝐵𝐶 = 𝑥 + 𝑦

2. 𝑚∠𝑅 = 90 − 𝑦, 𝑚∠𝐶 = 13, 𝑃𝑅 = 3𝑥 + 𝑦 − 1, 𝐴𝐶 = 32 − 𝑥

3. 𝑃𝑄 = 5𝑥 − 31, 𝑄𝑅 = −3𝑦 − 1, 𝐵𝐶 = 𝑥 + 1, 𝐴𝐵 = 9 − 𝑦

4. 𝑚∠𝐴 = 15𝑦 − 3, 𝑚∠𝑃 = 43 − 𝑥, 𝑃𝑄 = 11 − 𝑥, 𝐴𝐵 = 3𝑦 + 1

5. 𝐴𝐵 = 2𝑥 + 𝑦, 𝑃𝑄 = 7, 𝐵𝐶 = 11, 𝑄𝑅 = 4𝑥 + 𝑦

6. Δ𝑋𝑌𝑍 ≅ Δ𝑀𝑁𝑂, 𝑚∠𝑋 = 𝑥 + 10, 𝑚∠𝑀 = 𝑦 + 20, 𝑚∠𝑌 = 3𝑥, and 𝑚∠𝑁 = 𝑥 + 3𝑦. Find 𝑚∠𝑋 and

𝑚∠𝑌.

II. Indicate which triangles are congruent. Be sure to have the correspondence of the letters correct.

a. Δ𝐸𝑅𝐶 ≅ _______ b. E is the midpoint of 𝑇𝑃̅̅̅̅ c. Δ𝐵𝑂𝑊 ≅ _______

Why is 𝑅𝐶̅̅ ̅̅ ≅ 𝑅𝐶̅̅ ̅̅ ? Δ𝑆𝑃𝐸 ≅ _______ Why is ∠1 ≅ ∠2?

E C

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III. Coordinate Geometry

1. Graph each line on a coordinate plane. Identify two congruent triangle formed by the lines. Explain why

the triangles are congruent.

x=0, y = 0, x = 4, y = 2x – 4

2. Consider two triangles, Δ𝐴𝐵𝐶 and Δ𝐹𝐷𝐸, with vertices A = (0, 7), B = (-4, 0), C = (0, 0), D = (2, 3),

E = (2, -1), and F = (9, -1). Draw a diagram and explain why Δ𝐴𝐵𝐶 ≅ Δ𝐹𝐷𝐸.

IV. Solve.

1. The perimeter of ABCD is 85. Find the value of x. Is Δ𝐴𝐵𝐶 congruent to Δ𝐴𝐷𝐶? Explain.

2. Given: Δ𝑁𝐸𝑊 ≅ Δ𝐶𝐴𝑅

EN = 11

AR = 2x – 4y

NW = x + y

CA = 4x + y

EW = 10

Draw the triangles, solve for x and y, and find CR.

3x + 4 5x - 7

6x - 114x

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Introduction to Triangle ≅ Proof Ex 1) Given: 𝐴𝐷̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅

𝐴𝐶̅̅ ̅̅ ⊥ 𝐵𝐷̅̅ ̅̅ Prove: ΔABD ≅ ΔCBD Given Given Reflexive Prop ≅ Ex 2) Given: <E ≅ <H G is the midpoint of 𝐸𝐻̅̅ ̅̅ Prove: ΔGFE ≅ ΔGIH

Ex 3) Given: 𝐽𝐾̅̅ ̅// 𝑀𝐿̅̅ ̅̅

𝐽𝐾̅̅ ̅ ≅ 𝑀𝐿̅̅ ̅̅ Prove: ∡𝐽 ≅ ∡𝐿

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G is the midpoint of 𝐸𝐻̅̅ ̅̅

<FGE ≅ <IGH

𝐽𝐾̅̅ ̅// 𝑀𝐿̅̅ ̅̅ 𝐽𝐾̅̅ ̅ ≅ 𝑀𝐿̅̅ ̅̅

Δ ≅Δ

Reflexive Prop ≅

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Ex 4) Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ ⊥ 𝐵𝐷̅̅ ̅̅ Prove: ΔABD ≅ ΔCBD Use separate paper to complete the following.

Ex 5) Given: 𝐴𝐵̅̅ ̅̅ // 𝐶𝐷̅̅ ̅̅

𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ Prove: 𝐴𝐸̅̅ ̅̅ ≅ 𝐸𝐶̅̅ ̅̅ Ex 6) Given: 𝐼𝐺̅̅ ̅ bisects < FIJ 𝐼𝐹̅̅ ̅ ≅ 𝐼𝐻̅̅̅̅ Prove: ∡F ≅ ∡H Ex 7) Given: 𝐾𝐿̅̅ ̅̅ // 𝐽𝑀̅̅ ̅̅ 𝐾𝐽̅̅ ̅// 𝐿𝑀̅̅ ̅̅ Prove: 𝐾𝐽̅̅ ̅ ≅ 𝐿𝑀̅̅ ̅̅

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F HG

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Reflexive Prop ≅

Def of right Δ

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Triangle Proofs Practice Complete the following proofs using separate paper. Draw and mark each picture before writing the proof.

1. Given: BD⊥AC

AD̅̅ ̅̅ ≅ DC̅̅ ̅̅

Prove: ∠ABD≅∠CBD

2. Given: G is the midpoint of FH

𝐸𝐹̅̅ ̅̅ ≅ 𝐿𝐻̅̅ ̅̅

∠E≅∠L

Prove: EG≅LG

3. Given: 𝐶𝐷̅̅ ̅̅ bisects ∠𝐴𝐶𝐵

∠𝐴 ≅ ∠𝐵

Prove: 𝐴𝐷̅̅ ̅̅ ≅ 𝐷𝐵̅̅ ̅̅

4. Given: BD⊥AC

AB̅̅ ̅̅ ≅ BC̅̅̅̅

Prove: ∠ABD≅∠CBD

5. Given: 𝑃𝑅̅̅ ̅̅ ≅ 𝑄𝑆̅̅ ̅̅

∡P≅∡S

∡T≅V

Prove: 𝑇𝑅̅̅ ̅̅ ≅ 𝑄𝑉̅̅ ̅̅

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6. Given: 𝐵𝐷̅̅ ̅̅ bisects ∠𝐴𝐵𝐶

𝐵𝐴̅̅ ̅̅ ≅ 𝐶𝐵̅̅ ̅̅

Prove: ∡ADB ≅ ∡BDC

7. Given: G is the midpoint of FI̅ ∡F ≅ ∡I

Prove: 𝐸𝐹̅̅ ̅̅ ≅ 𝐼𝐻̅̅̅̅

8. Given: 𝐸𝐹̅̅ ̅̅ //𝐻𝐼̅̅̅̅ G is the midpoint of 𝐸𝐻̅̅ ̅̅

Prove: 𝐹𝐺̅̅ ̅̅ ≅ 𝐺𝐻̅̅ ̅̅

9. Given: 𝐽𝑀̅̅ ̅̅ //𝐿𝐾̅̅ ̅̅ ∡J ≅ ∡L

Prove: 𝐽𝐾̅̅ ̅ ≅ 𝑀𝐿̅̅ ̅̅

10. Given: 𝐽𝑀̅̅ ̅̅ //𝐿𝐾̅̅ ̅̅ 𝐽𝐾̅̅ ̅//𝐿𝑀̅̅ ̅̅

Prove: ∡J ≅ ∡L

B

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MORE PRACTICE WITH PROOF EX 1) GIVEN: 𝑀𝑄̅̅ ̅̅ ̅ ≅ 𝑃𝑅̅̅ ̅̅ , <M AND <P ARE RIGHT ANGLES. N IS THE MIDPOINT OF 𝑀𝑃̅̅̅̅̅ PROVE: <MQN ≅ <PRN

EX 2) GIVEN: 𝐴𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ <ADC ≅ < BCD

PROVE: 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅

EX 3) GIVEN: <I ≅ <G <1 ≅ <2

𝐽�̅� ≅ 𝐾𝐺̅̅ ̅̅ PROVE: <1 ≅ < 2

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EX 4) GIVEN: 𝐴𝐵̅̅ ̅̅ // 𝐶𝐷̅̅ ̅̅ 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ <AEB ≅ <DFC

PROVE: 𝐵𝐸̅̅ ̅̅ ≅ 𝐷𝐹̅̅ ̅̅

EX 5) GIVEN: 𝑃𝐼̅̅ ̅ ≅ 𝐷𝐼̅̅ ̅ 𝑅𝐼̅̅ ̅ ≅ 𝐸𝐼̅̅ ̅ PROVE: ∠𝑅 ≅ ∠𝐸

EX 6) GIVEN: 𝐾𝑀̅̅ ̅̅ ̅ ⊥ 𝐽�̅� 𝑀 IS THE MIDPOINT OF 𝐽�̅� PROVE: ∆𝐽𝐾𝑀 ≅ ∆𝐿𝐾𝑀

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Unit 4B Day 4: More Practice with Proof

1. Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

𝐴𝐵̅̅ ̅̅ ⊥ 𝐵𝐶̅̅ ̅̅

𝐶𝐷̅̅ ̅̅ ⊥ 𝐵𝐶̅̅ ̅̅

𝐴𝐸̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

Prove: ∡A ≅ ∡D

2. Given: 𝐺𝐾̅̅ ̅̅ ≅ 𝐻𝐿̅̅ ̅̅

𝐺𝐿̅̅̅̅ ≅ 𝐻𝐾̅̅ ̅̅

Prove: ∡K ≅ ∡L

3. Given: 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅

𝐴𝐸̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅

Prove: 𝐶𝐷̅̅ ̅̅ ≅ 𝐶𝐸̅̅ ̅̅

4. Given: <F and <H are right angles

G is the midpoint of 𝐹𝐻̅̅ ̅̅

𝐸𝐺̅̅ ̅̅ ≅ 𝐿𝐺̅̅̅̅

Prove: ∡E ≅ ∡L

5. Given: ∡1 ≅ ∡2

∡B ≅ ∡ECF

𝐵𝐷̅̅ ̅̅ ≅ 𝐶𝐹̅̅ ̅̅

Prove: 𝐴𝐷̅̅ ̅̅ ≅ 𝐸𝐹̅̅ ̅̅

B

C

A

D

E

F

G H

K L

C

A B

ED

21

A

B F

E

C D

25

6. Given: ∡B ≅ ∡C

𝐵𝐹̅̅ ̅̅ ≅ 𝐺𝐶̅̅ ̅̅

𝐵𝐷̅̅ ̅̅ ≅ 𝐸𝐶̅̅ ̅̅

Prove: ∡BDF ≅ ∡CEG

7. Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

𝐴𝐵̅̅ ̅̅ // 𝐶𝐷̅̅ ̅̅

𝐴𝐸̅̅ ̅̅ ≅ 𝐶𝐹̅̅ ̅̅

Prove: 𝐵𝐸̅̅ ̅̅ ≅ 𝐷𝐹̅̅ ̅̅

8. Given: ∠𝐷𝐴𝐿 ≅ ∠𝐵𝐶𝑀

𝐷𝐿̅̅ ̅̅ ≅ 𝑀𝐵̅̅ ̅̅̅

∡ALD and CMB are right angles

Prove: 𝐴𝐿̅̅̅̅ ≅ 𝐶𝑀̅̅̅̅̅

9. Given: 𝐹𝐼̅̅ ̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝐸𝐻̅̅ ̅̅

∡E ≅ ∡H

Prove: 𝐸𝐹̅̅ ̅̅ ≅ 𝐻𝐼̅̅̅̅

10. Given: 𝐹𝐼̅̅ ̅𝑎𝑛𝑑 𝐻𝐸̅̅ ̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟

Prove: ∡E ≅ ∡H

B C

A

F

DE

G

2

1

D

B C

A

E

F

2

1

B

D C

A

L

M

G

F

E

H

I

G

F

E

H

I

26

Isosceles Triangle investigation

1. In the box, draw an angle

and label the vertex C. This

will be your vertex angle.

Measure C.

2. Using point C as center,

swing an arc that intersects

both sides of C

3. Label the points of

intersection A and B.

Construct side AB. You

have constructed isosceles

ΔABC with base AB.

4. Measure sides AC and BC. What is the relationship between AC and BC?

5. Use your protractor to measure the base angles (A and B) of isosceles ΔABC.

6. Compare your results with the rest of the class. What relationship do you notice about the

base angles of each isosceles triangle?

Isosceles Triangle Theorem: If 2 sides of a triangle are congruent, then

_______________________________________________________________________________.

Isosceles Triangle Theorem Converse: If 2 angles of a triangle are congruent, then

______________________________________________________________________________.

1. If 𝐶𝑀̅̅̅̅̅ ≅ 𝐸𝑀̅̅̅̅̅, then _________ ≅ _________ by

_____________________________________________.

2. If ∠2 ≅ ∠3, then _________ ≅ _________ by

_____________________________________________.

431 2

OC E

M

R

27

Isosceles Triangle Practice 1. In triangle ABC, AB CB and mCBD = 124. Find the measure of A.

2. In isosceles triangle ABC, AB = AC. mC = 6x + 10 and mB = 3x + 40. Find the measure of the exterior angle at the vertex angle A.

3. Find the value of x: a.

b.

c.

28

5

43Y

X

Z

1 23 4

ZK

V

LO

21

P

A DB C

Practice with Isosceles Triangle Theorem and Converse Proofs

1. Given: 𝑌𝑋̅̅ ̅̅ ≅ 𝑋𝑍̅̅ ̅̅

Prove: ∡3 ≅ ∡5

2. Given: 𝐾𝑉̅̅ ̅̅ ≅ 𝑉𝑍̅̅̅̅

𝐾𝑂̅̅ ̅̅ ≅ 𝐿𝑍̅̅̅̅

Prove: ΔKVO ≅ ΔZVO

3. Given: ∡1 ≅ ∡2

𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

𝐴𝑃̅̅ ̅̅ ≅ 𝑃𝐷̅̅ ̅̅

Prove: ΔABP ≅ ΔDCP

29

T

P RI

AE

21G

J

K

M

50

x

x 54

63

11

10x

2x + 75x - 8

4040

F G

4x - 6

18

16

F

G H

x

98

x

74

12

10

10

A D

E

B C

Find the value of the variable or question mark. 1. 2. 3. 4. x = ________ x = ________ x = ________ x = ________ 5. 6. 7. 8. x = ________ x = ________ x = ________ x = ________ Complete the following using the diagram to the right. 9. a. If 𝐸𝐴̅̅ ̅̅ ≅ 𝐸𝐷̅̅ ̅̅ , then _______ _______.

b. If 𝐸𝐵̅̅ ̅̅ ≅ 𝐸𝐶̅̅ ̅̅ , then _______ _______.

c. If ΔEAD is an isosceles right triangle with right angle AED,

then the measure of A is ________.

10. Given: 𝑃𝑇̅̅̅̅ ≅ 𝑇𝑅̅̅ ̅̅

𝐸𝑃̅̅ ̅̅ ≅ 𝐴𝑅̅̅ ̅̅

I is the midpoint of 𝑃𝑅̅̅ ̅̅

Prove: 𝐸𝐼̅̅ ̅ ≅ 𝐴𝐼̅̅ ̅

11. Given: M is the midpoint of 𝐽𝐾̅̅ ̅

∠1≅∠2

Prove: JG ≅MK

30

DB C

A

Use your knowledge of congruent triangle proofs to complete the following flow proofs. 1. Given: 𝐴𝐷 ⃡ 𝑖𝑠 𝑡ℎ𝑒 ⊥ 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝐵𝐶̅̅ ̅̅

Prove: 𝐴𝐵 = 𝐴𝐶

𝐴𝐷 ⃡ 𝑖𝑠 𝑡ℎ𝑒 ⊥ 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝐵𝐶̅̅ ̅̅

𝐴𝐷 ⃡ ⊥ 𝐵𝐶̅̅ ̅̅

𝐷 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐶̅̅ ̅̅

Reflexive prop of ≅

∠𝐴𝐷𝐵 ≅ ∠𝐴𝐷𝐶 ∆𝐴𝐵𝐷≅ ∆𝐴𝐶𝐷

𝐴𝐵 = 𝐴𝐶

Theorem: If a point is on the perpendicular bisect of a segment,

it is __________________________________ from the _________________________

of the segment.

31

4

3

2

1

E

FB

D

A

C

Given: 𝐷 𝑖𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 ∠𝐴𝐵𝐶

𝐷𝐸̅̅ ̅̅ ⊥ 𝐵𝐴 , 𝐷𝐹̅̅ ̅̅ ⊥ 𝐵𝐶

Prove: 𝐷𝐸 = 𝐷𝐹

𝐷 𝑖𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 ∠𝐴𝐵𝐶

𝐷𝐸̅̅ ̅̅ ⊥ 𝐵𝐴

𝐷𝐹̅̅ ̅̅ ⊥ 𝐵𝐶

reflexive prop of ≅

Theorem: If a point is on the bisector of an angle, it is __________________________________

from the _________________________ of the angle.

32

BC D

A

P

NM

Q

L

O

PH

I

G

T

S

U

R

B

D

C

A

E

W

VK

F

J

A B

P C

Q

Let’s Make a Proof! You have spent lots of time in this unit writing proofs that someone else designed. Now it is your turn to make your own! Use the guidelines below to make up and do at least 2 proofs. Each of your 2 proofs should consist of: One of the given diagrams

The appropriate “given” and “prove” statements to set up your proof

The correctly completed proof using the diagram and “given” and “prove” statement you wrote

At least 3 vocabulary words from the given list (within the completed proof)

At least 3 of the rules from the postulates, property, and theorems list (within the completed

proof)

The second proof you write must use a different diagram, different vocabulary words, and different rules from the first one to meet the minimum requirements.

Diagram Choices: 1. 2. 3. 4. 5. 6. 7.

Vocabulary Terms: o Congruent segments

o Congruent angles

o Midpoint

o Segment bisector

o Angle bisector

o Perpendicular lines

o Perpendicular bisector

o Right angle

o Right triangle

Rules (Postulates, Properties, Theorems): o Vertical angles are congruent.

o Reflexive property of congruence

o All right angles are congruent.

o If ∥ lines are cut by a transv., corr. ∠𝑠 are ≅.

o If ∥ lines are cut by a transv., alt. int. ∠𝑠 are ≅.

o If ∥ lines are cut by a transv., alt. ext. ∠𝑠 are ≅.

o SSS

o SAS

o ASA

o AAS

o HL

o CPCTC

o Isosceles Triangle Theorem

o Isosceles Triangle Theorem Converse

33

Test Review 1. Given BED BUG, find x and BU if BE = 6x+16, UG=10x-2 and ED = 6x+6.

x=__________

BU=________________

2. In ∆ABC, AB, AC = 6x-5, BC = 3x+13, and AB = 4x+7. Find x and the length of the base.

x

___________

base =

___________

3. In an isosceles triangle, a vertex angle measures 36⁰. What is the measure of each base angle? 4. Decide whether it is possible to prove the triangles are congruent. If yes, then state which

congruence postulate you would use.

a. b. c. d. 5. If ABCPQR and C=2x+2, R=3x-18, find the value of x. 6. ∆XYZ ∆JKL. IfY=14-x, K=2x+50 and L=-4x find the Zm . 7. IfABCLMN, and AB=4x-y, LM =2x-2y, BC = x-3y and MN =21 find the value of x and y.

34

8. Suppose ∆ABC ∆EFG. For each of the following, name the corresponding part.

A : ______________ BCA : ______________ AC : _______________

9.

10. In the given triangles, ABCXYZ Which two statements identify corresponding congruent parts for these triangles?

A. B XY and CY

B. AB YZ and CX

C. BC XY and AY

D. BC YZ and AX

11. BOY DEA. YB = 4x+8, AD = 60, AE = 80, AND DE = 40. Find x.

12. EXC STP. C = 3x+12, T = 81, and X = 12x – 15 . Find m C . 13. MNODEF. Calculate the value for x, y, and z. 𝑚∠𝑀 = 50 x = ________ 𝑚∠𝑁 = 60 𝑚∠𝑂 = 70 y = ________ 𝑚∠𝐷 = (2𝑥 − 20)

𝑚∠𝐸 = (1

2𝑦 + 10) z = ________

𝑚∠𝐹 = (10 + 𝑧) 14. The figure shows . ABCEDC. C is the midpoint of BD and AE. What

reason would you use to prove the triangles are congruent, if any?

If the triangles are congruent, complete the statement: A _____

35

A

B C RQ

P P

Q RCB

A P

Q RCB

A

RQ

PA

B C CB

A P

Q R

x

40

8

10

10

2x - 10

2x + 1 x + 580

20

15. If ∆JKM ∆RST, how do you know JK RS? A. Definition of a line segment C. SSS Postulate B. CPCTC D. SAS Postulate 16-20 Proving Triangles are Congruent (SSS, SAS, ASA, AAS, HL) For each of the following, give

the reason for triangle congruence.

16. 17. 18.

19. 20.

21& 22. Solve for x 21 22.