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Splash Screen. 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

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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

Find m1.

A.115

B.105

C.75

D.65

Find m2.

A.75

B.72

C.57

D.40

Find m3.

A.75

B.72

C.57

D.40

Find m4.

A.18

B.28

C.50

D.75

Find m5.

A.70

B.90

C.122

D.140

5-Minute Check 5

One 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 6

Content Standards

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.

CCSS

You identified and used congruent angles.

• Name and use corresponding parts of congruent polygons.

• Prove triangles congruent using the definition of congruence.

Then/Now

• congruent

• congruent polygons

• corresponding parts

Concept 1

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 1

A.

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 1

Use Corresponding Parts of Congruent Triangles

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

OPCPCTC

mO=mPDefinition of congruence

6y – 14=40Substitution

Example 2

CPCTC

Use Corresponding Parts of Congruent Triangles

y=9Divide each side by 6.

NG=ITDefinition of congruence

x – 2y=7.5Substitution

x – 2(9)=7.5y = 9

x – 18=7.5Simplify.

Answer:x = 25.5, y = 9

Example 2

In the diagram, ΔFHJ 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

Concept 2

Use the Third Angles Theorem

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 mIJK = 72, find mJIH.

ΔJIK  ΔJIH Congruent Triangles

mIJK + mIKJ + mJIK=180Triangle Angle-SumTheorem

Example 3

Use the Third Angles Theorem

mIJK + mIJK + mJIK =180Substitution

72 + 72 + mJIK =180Substitution

144 + mJIK =180Simplify.

mJIK =36Subtract 144 fromeach side.

mJIH =36Third Angles Theorem

Example 3

TILES A drawing of a tile contains a series of triangles, rectangles, squares, and a circle. If ΔKLM  ΔNJL, KLM  KML,and mKML = 47.5, find mLNJ.

A.85

B.45

C.47.5

D.95

Example 3

Prove That Two Triangles are Congruent

Write a two-column proof.

Prove:ΔLMNΔPON

Example 4

StatementsReasons

1. Given

1.

2. LNM  PNO

2. Vertical Angles Theorem

3.M  O

3. Third Angles Theorem

4.ΔLMNΔPON

4. CPCTC

Prove That Two Triangles are Congruent

Proof:

Example 4

Statements

Reasons

1. Given

1.

2. Reflexive Property ofCongruence

2.

3.Q  O, NPQ  PNO

3. Given

4. _________________

4.QNP  ONP

?

5.ΔQNPΔOPN

5. Definition of Congruent Polygons

Find the missing information in the following proof.

Prove:ΔQNPΔOPN

Proof:

Example 4

A.CPCTC

B.Vertical Angles Theorem

C.Third Angles Theorem

D.Definition of Congruent Angles