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BC DA , BC AD. ABC CDA. STATEMENTS. REASONS. S. BC DA. Given. Given. BC AD. BCA DAC. A. Alternate Interior Angles Theorem. S. AC CA. Reflexive Property of Congruence. EXAMPLE 1. Use the SAS Congruence Postulate. Write a proof. GIVEN. PROVE. EXAMPLE 1.

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EXAMPLE 1

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Example 1

BC DA,BC AD

ABCCDA

STATEMENTS

REASONS

S

BC DA

Given

Given

BC AD

BCADAC

A

Alternate Interior Angles Theorem

S

ACCA

Reflexive Property of Congruence

EXAMPLE 1

Use the SAS Congruence Postulate

Write a proof.

GIVEN

PROVE


Example 1

EXAMPLE 1

Use the SAS Congruence Postulate

STATEMENTS

REASONS

ABCCDA

SAS Congruence Postulate


Example 1

Because they are vertical angles, PMQRMS. All points on a circle are the same distance from the center, so MP, MQ, MR, and MSare all equal.

ANSWER

MRSand MPQ are congruent by the SAS Congruence Postulate.

EXAMPLE 2

Use SAS and properties of shapes

In the diagram, QSand RPpass through the center Mof the circle. What can you conclude about MRSand MPQ?

SOLUTION


Example 1

Prove that

SVRUVR

STATEMENTS

REASONS

SV VU

Given

SVRRVU

Definition of line

Reflexive Property of Congruence

RVVR

SVRUVR

SAS Congruence Postulate

for Examples 1 and 2

GUIDED PRACTICE

In the diagram, ABCDis a square with four congruent sides and four right angles. R, S, T, and Uare the midpoints of the sides of ABCD. Also, RT SUand .

SU VU


Example 1

STATEMENTS

REASONS

Given

BS DU

Definition of line

RBSTDU

Given

RSUT

SAS Congruence Postulate

BSRDUT

for Examples 1 and 2

GUIDED PRACTICE

BSRDUT

Prove that


Example 1

GIVEN

WY XZ,WZ ZY, XY ZY

WYZXZY

PROVE

Redraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram.

EXAMPLE 3

Use the Hypotenuse-Leg Congruence Theorem

Write a proof.

SOLUTION


Example 1

STATEMENTS

REASONS

H

WY XZ

Given

WZ ZY, XY ZY

Given

Definition of lines

Z andY are right angles

Definition of a right triangle

WYZand XZY are right triangles.

ZY YZ

L

Reflexive Property of Congruence

WYZXZY

HL Congruence Theorem

EXAMPLE 3

Use the Hypotenuse-Leg Congruence Theorem


Example 1

EXAMPLE 4

Choose a postulate or theorem

Sign Making

You are making a canvas sign to hang on the triangular wall over the door to the barn shown in the picture. You think you can use two identical triangular sheets of canvas. You knowthatRP QS andPQ PS . What postulate or theorem can you use to conclude that

PQR PSR?


Example 1

You are given that PQ PS. By the Reflexive Property, RP RP. By the definition of perpendicular lines, both

RPQ and RPSare right angles, so they are congruent. So, two sides and their included angle are congruent.

ANSWER

You can use the SAS Congruence Postulate to conclude that .

PQR PSR

EXAMPLE 4

Choose a postulate or theorem

SOLUTION


Example 1

Redraw ACBand DBCside by side with corresponding parts in the same position.

for Examples 3 and 4

GUIDED PRACTICE

Use the diagram at the right.


Example 1

STATEMENTS

REASONS

H

AB DC

Given

AC BC, DB BC

Given

Definition of lines

C B

Definition of a right triangle

ACBand DBC are right triangles.

for Examples 3 and 4

GUIDED PRACTICE

Use the diagram at the right.

Use the information in the diagram to prove that

ACB DBC


Example 1

BC CB

L

Reflexive Property of Congruence

ACBDBC

HL Congruence Theorem

for Examples 3 and 4

GUIDED PRACTICE

STATEMENTS

REASONS


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