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Five-Minute Check (over Lesson 4–2)

CCSS

Then/Now

New Vocabulary

Key Concept: Definition of Congruent Polygons

Example 1: Identify Corresponding Congruent Parts

Example 2: Use Corresponding Parts of Congruent Triangles

Theorem 4.3: Third Angles Theorem

Example 3: Real-World Example: Use the Third Angles Theorem

Example 4: Prove that Two Triangles are Congruent

Theorem 4.4: Properties of Triangle Congruence

Lesson MenuOne angle in an isosceles triangle has a measure of 80°. What is the measure of one of the other two angles?

A. 35

B. 40

C. 50

D. 100

5-Minute Check 6Content Standards What is the measure of one of the other two angles?

G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Mathematical Practices

6 Attend to precision.

3 Construct viable arguments and critique the reasoning of others.

CCSSYou identified and used congruent angles. What is the measure of one of the other two angles?

- Name and use corresponding parts of congruent polygons.

- Prove triangles congruent using the definition of congruence.

- congruent What is the measure of one of the other two angles?
- congruent polygons
- corresponding parts

Concept 1 What is the measure of one of the other two angles?

Angles: What is the measure of one of the other two angles?

Sides:

Identify Corresponding Congruent Parts

Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.

Answer: All corresponding parts of the two polygons are congruent. Therefore, ABCDE RTPSQ.

Example 1A. What is the measure of one of the other two angles?

B.

C.

D.

The support beams on the fence form congruent triangles. In the figure ΔABC ΔDEF,which of the following congruence statements correctly identifies corresponding angles or sides?

Example 1Use Corresponding Parts of Congruent Triangles What is the measure of one of the other two angles?

In the diagram, ΔITP ΔNGO. Find the values of x and y.

O P CPCTC

mO = mP Definition of congruence

6y – 14 = 40 Substitution

Example 2CPCTC What is the measure of one of the other two angles?

Use Corresponding Parts of Congruent Triangles

6y = 54Add 14 to each side.

y= 9Divide each side by 6.

NG= ITDefinition of congruence

x – 2y = 7.5 Substitution

x – 2(9) = 7.5 y = 9

x – 18 = 7.5 Simplify.

x= 25.5Add 18 to each side.

Answer:x = 25.5, y = 9

Example 2In the diagram, Δ What is the measure of one of the other two angles?FHJ ΔHFG. Find the values of x and y.

A.x = 4.5, y = 2.75

B.x = 2.75, y = 4.5

C.x = 1.8, y = 19

D.x = 4.5, y = 5.5

Example 2Concept 2 What is the measure of one of the other two angles?

Use the Third Angles Theorem What is the measure of one of the other two angles?

ARCHITECTURE A drawing of a tower’s roof is composed of congruent triangles all converging at a point at the top. If IJK IKJ and mIJK = 72, find mJIH.

ΔJIK ΔJIH Congruent Triangles

mIJK + mIKJ + mJIK = 180 Triangle Angle-Sum Theorem

Example 3Use the Third Angles Theorem What is the measure of one of the other two angles?

mIJK + mIJK + mJIK = 180 Substitution

72 + 72 + mJIK = 180 Substitution

144 + mJIK = 180 Simplify.

mJIK = 36 Subtract 144 from each side.

mJIH = 36 Third Angles Theorem

Answer:mJIH = 36

Example 3TILES What is the measure of one of the other two angles? A drawing of a tile contains a series of triangles, rectangles, squares, and a circle. If ΔKLM ΔNJL, KLM KML,and mKML = 47.5, find mLNJ.

A. 85

B. 45

C. 47.5

D. 95

Example 3Prove That Two Triangles are Congruent What is the measure of one of the other two angles?

Write a two-column proof.

Prove:ΔLMNΔPON

Example 4Statements Reasons What is the measure of one of the other two angles?

1. Given

1.

2. LNM PNO

2. Vertical Angles Theorem

3. M O

3. Third Angles Theorem

4. ΔLMNΔPON

4. CPCTC

Prove That Two Triangles are Congruent

Proof:

Example 4Statements What is the measure of one of the other two angles?

Reasons

1. Given

1.

2. Reflexive Property of Congruence

2.

3.Q O, NPQ PNO

3. Given

4. _________________

4.QNP ONP

?

5.ΔQNPΔOPN

5. Definition of Congruent Polygons

Find the missing information in the following proof.

Prove:ΔQNPΔOPN

Proof:

Example 4A. What is the measure of one of the other two angles? CPCTC

B. Vertical Angles Theorem

C. Third Angles Theorem

D. Definition of Congruent Angles

Example 4Concept 3 What is the measure of one of the other two angles?

End of the Lesson What is the measure of one of the other two angles?

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