4.2 Apply Congruence and Triangles

• You will identify congruent figures.

• Essential Question: What are congruent figures?

Tell students they will learn how to answer this question by studying the definition of congruent figures

Warm-Up ExercisesEXAMPLE 1 Identify congruent parts

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

SOLUTION

The diagram indicates that JKL TSR.

Corresponding sides JK TS, KL SR, LJ RT

Corresponding angles J T, ∠K S, L R

Warm-Up ExercisesEXAMPLE 2 Use properties of congruent figures

In the diagram, DEFG SPQR.

Find the value of x.a.

b. Find the value of y.

SOLUTION

You know that FG QR.a.

FG = QR

12 = 2x – 4

16 = 2x

8 = x

Warm-Up ExercisesEXAMPLE 2 Use properties of congruent figures

b. You know that ∠F Q.

m F = m Q

68 o = (6y + x)o

68 = 6y + 8

10 = y

Warm-Up ExercisesEXAMPLE 3 Show that figures are congruent

PAINTING

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.

SOLUTION

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 .

Warm-Up Exercises

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

EXAMPLE 3 Show that figures 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.

Warm-Up ExercisesGUIDED PRACTICE for Examples 1, 2, and 3

1. Identify all pairs of congruent corresponding parts.

SOLUTION

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.

Warm-Up ExercisesGUIDED PRACTICE for Examples 1, 2, and 3

In the diagram at the right, ABGH CDEF.

SOLUTION

2. Find the value of x and find m H.

(b) You know that H Fm H m F =105°

(a) You know that H F (4x+ 5)° = 105°

4x = 100 x = 25

Warm-Up ExercisesGUIDED PRACTICE for Examples 1, 2, and 3

In the diagram at the right, ABGH CDEF.

3. Show that PTS RTQ.

SOLUTION

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.

Warm-Up ExercisesEXAMPLE 4 Use the Third Angles Theorem

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° .

Warm-Up ExercisesEXAMPLE 5 Prove that triangles are congruent

Plan for Proof

AC AC.

a. Use the Reflexive Property to show that

b. Use the Third Angles Theorem to show that

B D

Write a proof.

GIVEN AD CB, DC AB

ACD CAB, CAD ACB

PROVE ACD CAB

Warm-Up ExercisesEXAMPLE 5 Prove that triangles are congruent

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 of5.

Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5

SOLUTION

4. DCN.In the diagram, what is m

CDN NSR, DNC SNR then the third angles are also congruent NRS DCN = 75°

Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5

By the definition of congruence, whatadditional information is needed toknow that

5.

NDC NSR.

ANSWER

DC RS and DN SN

Warm-Up ExercisesDaily Homework Quiz

CAANSWER

3. EDF ?

In the diagram, ABC DEF. Complete the statement.

BACANSWER

60°ANSWER

2. FD ?

m A = ?1.

Warm-Up ExercisesDaily Homework Quiz

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

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 Segs.

s

ANSWER

• You will identify congruent figures.

• Essential Question: What are congruent figures?• Triangles can be proved

congruent by showing that all 3 pairs of corresponding sides and all 3 pairs of corresponding anglesare congruent.• If two angles of one triangle are congruent to two angles of another, then the third angles are congruent.

Congruent figures are figures that have exactly the same size and shape.