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

Over Lesson 4–2

A. 115

B. 105

C. 75

D. 65

Find m1.

Over Lesson 4–2

A. 75

B. 72

C. 57

D. 40

Find m2.

Over Lesson 4–2

A. 75

B. 72

C. 57

D. 40

Find m3.

Over Lesson 4–2

A. 18

B. 28

C. 50

D. 75

Find m4.

Over Lesson 4–2

A. 70

B. 90

C. 122

D. 140

Find m5.

Over Lesson 4–2

A. 35

B. 40

C. 50

D. 100

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

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.

You identified and used congruent angles.

• Name and use corresponding parts of congruent polygons.

• Prove triangles congruent using the definition of congruence.

• congruent

• congruent polygons

• corresponding parts

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.

Sides:

Angles:

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?

A.

B.

C.

D.

Use Corresponding Parts of Congruent Triangles

O P CPCTC

mO = mP Definition of congruence

6y – 14 = 40 Substitution

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

Use Corresponding Parts of Congruent Triangles

6y = 54 Add 14 to each side.

y = 9 Divide each side by 6.

NG = IT Definition of congruence

x – 2y = 7.5 Substitution

x – 2(9) = 7.5 y = 9

x – 18 = 7.5 Simplify.

x = 25.5 Add 18 to each side.

CPCTC

Answer: x = 25.5, y = 9

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

In the diagram, ΔFHJ ΔHFG. Find the values of x and y.

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 mIJK = 72, find mJIH.

mIJK + mIKJ + mJIK = 180 Triangle Angle-SumTheorem

ΔJIK ΔJIH Congruent Triangles

Use the Third Angles Theorem

mIJK + mIJK + mJIK = 180 Substitution

72 + 72 + mJIK = 180 Substitution

144 + mJIK = 180 Simplify.

mJIK = 36 Subtract 144 fromeach side.

mJIH = 36 Third Angles Theorem

Answer: mJIH = 36

A. 85

B. 45

C. 47.5

D. 95

TILES 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.

Prove That Two Triangles are Congruent

Write a two-column proof.

Prove: ΔLMN ΔPON

Prove That Two Triangles are Congruent

2. LNM PNO 2. Vertical Angles Theorem

Proof:

Statements Reasons

3. M O

3. Third Angles Theorem

4. ΔLMN ΔPON

4. CPCTC

1. Given1.

Find the missing information in the following proof.

Prove: ΔQNP ΔOPN

Proof:ReasonsStatements

3. Q O, NPQ PNO 3. Given

5. Definition of Congruent Polygons5. ΔQNP ΔOPN

4. _________________4. QNP ONP ?

2. 2. Reflexive Property ofCongruence

1. 1. Given

A. CPCTC

B. Vertical Angles Theorem

C. Third Angles Theorem

D. Definition of Congruent Angles