4.9 Prove Triangles Congruent by SAS

Side-Angle-Side (SAS) Congruence PostulateIf two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

_____,RS Side If UVand _____,R Angle U

_____,RT Side UW______ RSTthen UVW

S

R T

V

U W

4.9 Prove Triangles Congruent by SAS

Example 1 Use the SAS Congruence PostulateUse the SAS Congruence Postulate

RST.LMN that Show

M

N

L

R

S

T

SolutionIt is given that ,STMN ,RSLM and __________.

So, by the ____________________________,

._______LMN

SM SAS Congruence Postulate

RST

4.9 Prove Triangles Congruent by SAS

Hypotenuse-Leg Congruence TheoremIf the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are ___________.congruent

B

A C

E

D F

4.9 Prove Triangles Congruent by SAS

Example 2 Use the HL Congruence TheoremUse the HL Congruence TheoremWrite a proof.Given: ,ECAC

,BDAB ,BDED

.BD ofbisector a is ACProve: EDCABC B

A

C

D

E

Statements Reasons

ECAC .1 _______ .1,BDAB .2 _______ .2

BDED

GivenGiven

4.9 Prove Triangles Congruent by SAS

Example 2 Use the HL Congruence TheoremUse the HL Congruence TheoremWrite a proof.Given: ,ECAC

,BDAB ,BDED

.BD ofbisector a is ACProve: EDCABC B

A

C

D

E

Statements Reasons

are D and B .3 lines of sDefinition .3 right angles ._____________

EDC and ABC .4 .___________are

_____ a of sDefinition .4__________right triangles

right

triangle

4.9 Prove Triangles Congruent by SAS

Example 2 Use the HL Congruence TheoremUse the HL Congruence TheoremWrite a proof.Given: ,ECAC

,BDAB ,BDED

.BD ofbisector a is ACProve: EDCABC B

A

C

D

E

Statements Reasons

bisector a is AC .5 ________ .5 Given.BD ofDCBC .6

EDCABC 7. bisectorsegment of sDefinition .6

_______________________ 7. HL Congruence Theorem

4.9 Prove Triangles Congruent by SASCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 1. Decide whether enough information is given to

prove ABC LMN using the SAS Congruence Postulate.

A

B C

N M

L,LMAB MNBC MB

Sides:

Included angle:

Therefore there is enough information

4.9 Prove Triangles Congruent by SASCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 2. Write a proof. P

S

Q

R

Given: ,QSPR ,SRPS SRQR

Prove: QSRPRS

Statements Reasons

QSPR .1 _______ .1SRQR ,SRPS .2 _______ .2

GivenGiven

are R and S .3 _____________________ .3 Definition of perp. lines.anglesright QSR and PRS .4

.nglesright tria are______________________ .4 Definition of a right triangle

4.9 Prove Triangles Congruent by SASCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 2. Write a proof. P

S

Q

R

Given: ,QSPR ,SRPS SRQR

Prove: QSRPRS

Statements Reasons

RSSR .5 _______________________ .5

QSRPRS .6 _____________________ .6Reflexive Property of Cong.

HL Congruence Theorem

4.9 Prove Triangles Congruent by SAS

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