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4.2 Apply Congruence and Triangles

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- 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

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

The diagram indicates that

JKL TSR.

Corresponding angles

J T,∠ K S,L R

Corresponding sides

JK TS,KL SR,LJ RT

EXAMPLE 1

Identify congruent parts

SOLUTION

In the diagram, DEFG SPQR.

Find the value of x.

Find the value of y.

You know that FG QR.

FG

= QR

= 2x – 4

12

16

= 2x

8

= x

EXAMPLE 2

Use properties of congruent figures

SOLUTION

You know that∠ F Q.

m F

= mQ

68

= 6y + 8

68

o

= (6y + x)

o

10

= y

EXAMPLE 2

Use properties of congruent figures

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.

From the diagram, A Cand D Bbecause all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB DC.

EXAMPLE 3

Show that figures are congruent

SOLUTION

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

EXAMPLE 3

Show that figures are congruent

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

In the diagram at the right, ABGH CDEF.

Identify all pairs of congruent corresponding parts.

AB CD, BG DE,

GH FE, HA FC

A C, B D,

G E, H F.

for Examples 1, 2, and 3

GUIDED PRACTICE

SOLUTION

Corresponding sides:

Corresponding angles:

In the diagram at the right, ABGH CDEF.

2.

Find the value ofxand findm H.

(a)

You know that H F

(4x+ 5)° = 105°

4x = 100

x = 25

(b)

You know that H F

m H m F =105°

for Examples 1, 2, and 3

GUIDED PRACTICE

SOLUTION

In the diagram at the right, ABGH CDEF.

3.

Show thatPTS 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.

for Examples 1, 2, and 3

GUIDED PRACTICE

FindmBDC.

SOLUTION

AB and ADCBCD, so by the Third Angles Theorem, ACDBDC. By the Triangle Sum Theorem, m ACD = 180°– 45° – 30°= 105°.

So, mACD =mBDC = 105° by the definition of congruent angles.

ANSWER

EXAMPLE 4

Use the Third Angles Theorem

Write a proof.

GIVEN

AD CB,DC AB

CADACB

ACDCAB,

ACDCAB

PROVE

Use the Reflexive Property to show that

AC AC.

Use the Third Angles Theorem to show that

BD

EXAMPLE 5

Prove that triangles are congruent

Plan for Proof

STATEMENTS

REASONS

,DC BA

AD CB

Given

Reflexive Property of Congruence

AC AC.

ACDCAB,

Given

CADACB

BD

Third Angles Theorem

ACDCAB

Definition of

EXAMPLE 5

Prove that triangles are congruent

Plan in Action

In the diagram, what is m

DCN.

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

for Examples 4 and 5

GUIDED PRACTICE

SOLUTION

By the definition of congruence, what

additional information is needed to

know that

NDC NSR.

ANSWER

DC RS and DN SN

for Examples 4 and 5

GUIDED PRACTICE

In the diagram, ABCDEF. Complete the statement.

1.

m A = ?

ANSWER

60°

2.

FD ?

ANSWER

CA

3.

EDF ?

ANSWER

BAC

Daily Homework Quiz

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

4.

s

ANSWER

WXZ YXZ; The diagram tell us that W Y and WZX YZX.

WXZ YXZ by the Third Thm. From the diagram WX YX and WZYZ, and XZ XZ by Refl. Prop.

Of Segs.

Daily Homework Quiz

• Triangles can be proved congruent by showing that all 3 pairs of corresponding sides and all 3 pairs of corresponding angles

are congruent.

• If two angles of one triangle are congruent to two angles of another, then the third angles are congruent.

- You will identify congruent figures.

- Essential Question: What are congruent figures?

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