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Warm Up

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Prove Triangles Congruent by SAS and HL

Warm Up

Lesson Presentation

Lesson Quiz

Given: DF bisects CE, DC DE

Prove: ∆CDF ∆EDF

C

F

D

E

ANSWER

It is given thatDC DEand DFbisects CE.CF EF by the def. of bisector.DF DF by the Refl. Prop. of Segs. So ∆CDF∆EDFby the SSS Post.

Warm-Up

BC DA,BC AD

ABCCDA

ABCCDA

STATEMENTS

REASONS

Given

S

BC DA

Given

BC AD

BCADAC

A

Alternate Interior Angles Theorem

S

ACCA

Reflexive Property of Congruence

5.

SAS Congruence Postulate

5.

Example 1

Write a proof.

GIVEN:

PROVE:

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

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

SOLUTION

Prove that

SVRUVR

ANSWER

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

BSRDUT

Prove that

ANSWER

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

Guided Practice

WY XZ,WZ ZY, XY ZY

GIVEN:

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

Write a proof.

SOLUTION

STATEMENTS

REASONS

Given

WY XZ

H

Given

WZ ZY, XY ZY

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

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 and PQ PS. What postulate or theorem can you use to conclude that

PQRPSR?

Sign Making

Example 4

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 .

PQRPSR

Example 4

SOLUTION

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

ANSWER

Guided Practice

Use the diagram at the right.

Use the information in the diagram to prove that

ACBDBC

ANSWER

Guided Practice

Use the diagram at the right.

1.

ABE,CBD

ANSWER

SAS Post.

Lesson Quiz

Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.

2.

FGH,HJK

ANSWER

HL Thm.

Lesson Quiz

Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.

State a third congruence that would allow you to prove RSTXYZ by the SAS Congruence postulate.

3.

ST YZ, RS XY

ANSWER

SY.

Lesson Quiz

State a third congruence that would allow you to prove RSTXYZ by the SAS Congruence postulate.

4.

T Z, RT XZ

ANSWER

STYZ .

Lesson Quiz