4.24.2 Apply Congruence and TrianglesBell Thinger

1. When are two angles congruent?

ANSWER when they have the same measure

ANSWER 45º

2. In ∆ABC, if m A = 64º and m B = 71º, what is m C?

ANSWER Transitive Property

3. What property of angle congruence is illustrated by this statement? If A ≅ B and B ≅ C, then A ≅

C.

4.2

4.2Example 1

Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts.

SOLUTION

The diagram indicates that JKL ≅ TSR.

Corresponding angles ∠J ≅ ∠T, ∠K ≅ ∠S, ∠L ≅ ∠R

Corresponding sides JK ≅ TS, KL ≅ SR, LJ ≅ RT

4.2Example 2

In the diagram, DEFG ≅ SPQR.

Find the value of x.a.

SOLUTION

FG = QR

12 = 2x – 4

16 = 2x

8 = x

You know that FG ≅ QR.a.

4.2Example 2

In the diagram, DEFG ≅ SPQR.

SOLUTION

b. Find the value of y.

b. You know that ∠F ≅ ∠Q.

m∠F = m∠Q

68 o = (6y + x)o

68 = 6y + 8

10 = y

4.2Example 3

SOLUTION

If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape?Explain.

PAINTING

From the diagram, ∠A ≅ ∠C and ∠D ≅ ∠B because all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB || DC .

4.2Example 3

Then, ∠1 ≅ ∠4 and ∠2 ≅ ∠3 by the Alternate Interior Angles Theorem. So, all pairs of corresponding angles are congruent.

The diagram shows AJ ≅ CK , KD ≅ JB , and DA ≅ BC . By the Reflexive Property, JK ≅ KJ . All corresponding parts are congruent, so AJKD ≅ CKJB.

4.2Guided Practice

1. Identify all pairs of congruent corresponding parts.

ANSWER

Corresponding sides: AB ≅ CD, BG ≅ DE, GH ≅ FE, HA ≅ FC

Corresponding angles: ∠A ≅ ∠C, ∠B ≅ ∠D, ∠G ≅ ∠E, ∠H ≅ ∠F.

In the diagram at the right, ABGH ≅ CDEF.

4.2

3. Show that PTS ≅ RTQ.

ANSWER

All of the corresponding parts of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternate Interior Angle theorem.

Guided Practice

4.2

In the diagram at the right, ABGH ≅ CDEF.

2. Find the value of x and find m∠H.

ANSWER 25, 105°

Guided Practice

4.2

4.2Example 4

Find m∠BDC.

So, m∠ACD = m∠BDC = 105° by the definition of congruent angles.

ANSWER

SOLUTION

∠A ≅∠B and ∠ADC ≅∠BCD, so by the Third Angles Theorem, ∠ACD ≅∠BDC. By the Triangle Sum Theorem, m∠ACD = 180° – 45° – 30° = 105° .

4.2Example 5

Plan for Proof

b. Use the Third Angles Theorem to show that ∠B ≅ ∠D.

Write a proof.

PROVE: ACD ≅ CAB

GIVEN: AD ≅ CB, DC ≅ BA, ∠ACD ≅ ∠CAB, ∠CAD ≅ ∠ACB

a. Use the Reflexive Property to show that AC ≅ AC.

4.2Example 5

Plan in Action

1. Given

2. Reflexive Property of Congruence

STATEMENTS REASONS

3. Given

4. Third Angles Theorem

1. AD ≅ CB , DC ≅ BA

2. a. AC ≅ AC.

3. ∠ACD ≅ ∠CAB,∠CAD ≅ ∠ACB

4. b. ∠B ≅ ∠D

5. ACD ≅ CAB Definition of ≅ figures5.

4.2

4. In the diagram, what is m∠DCN.

ANSWER 75°

Guided Practice

4.2

By the definition of congruence, whatadditional information is needed toknow that NDC ≅ NSR.

5.

Guided Practice

ANSWER

DC ≅ RS and DN ≅ SN

4.2

4.2Exit Slip

CAANSWER

3. EDF ≅ ?

In the diagram, ABC ≅ DEF. Complete each statement.

BACANSWER

60°ANSWER

2. FD ≅ ?

m A = ?1.

4.2

4. Write a congruence statement for the two small triangles. Explain your reasoning.

Exit Slip

WXZ ≅ YXZ; The diagram tell us that W ≅ Y and WZX ≅ YZX. WXZ ≅ YXZ by the Third Thm. From the diagram WX ≅ YX and WZ ≅ YZ, and XZ ≅ XZ by Refl. Prop. Of ≅.

ANSWER

4.2

Homework

Pg 238-241#5, 15, 16, 19, 20