Warm Up: Tell whether it is possible to draw each triangle.

1. Acute scalene triangle2. Obtuse equilateral triangle3. Right isosceles triangle4. Scalene equiangular triangle5. Right scalene triangle

4.3 Congruent Triangles

4.3 – Congruent Triangles

•Two geometric figures are congruent if they have exactly the same size and shape.

•When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.

RPCA

QRBC

PQAB

A

B

CP

Q

R

Ex 1: Illustrate the two triangles.

What angles corresponding to what angles?

Ex 2: ABCD is to HGFE, find x and y.A B

DD CC

F E

GG HH

9191oo

8686oo

9cm(5y-12)(5y-12)oo

(4x-3)cm(4x-3)cm113113oo

Third Angles Theorem

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

Properties of Congruent Triangles•Reflexive Property: Every triangle is congruent to

itself.

•Symmetric Property of Congruent Triangles: If ABC = DEF, then DEF ABC̃ =̃

•Transitive Property of Congruent Triangles: If ABC DEF and DEF JKL, then =̃ =̃

ABC JKL. =̃

Ex 4:Ex 4: GivenGiven: seg RP : seg RP seg MN, seg PQ seg MN, seg PQ seg NQ, seg NQ, seg RQ seg RQ seg MQ, m seg MQ, mP = 92P = 92oo and m and mN is 92N is 92oo..

ProveProve: : ΔΔRQP RQP ΔΔMQNMQN

R

P

Q

N

M9292oo

9292oo

Section 4.4Proving Triangles Congruent: SSS and SAS

Using the SSS PostulateSide side side postulate: If three sides of one triangle are congruent to three

sides of another triangle, the the two triangles are congruent.

Example 1•Prove the triangles

congruent.

A

B

C

D

Given : CADB,CDABProve : CABBDC

Statement Reason

1.

2.

3.

Use the SSS Congruence Postulate to show that

ABC FGH.

Using the SAS PostulateSide angle side postulate: If two sides and the included angle of one triangle

are congruent to two sided and the included angle of another triangle, then the two triangles are congruent.

Example 2•Prove the triangles

congruent.

A B

CD

Given : AB ||CD & ABCDProve : ABDCDB

Statement Reason

1.

2.

3.

4.

Section 4.5

Proving Triangles Congruent: ASA and AAS

Using the ASA Postulate

• Angle side angle postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the two triangles are congruent.

Example 1

•Prove the triangles congruent.

ONPMNPOPNMPNONPMNP

:Prove & :Given

Statement Reason

M

N

O

P

M

N

O

P

N

P

Using the AAS Theorem

• Angle angle side theorem: If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent.

Example 2

•Prove the triangles congruent.

STRPQRSPSTP

:Prove & Q :Given

Statements Reasons

P

Q

RS

T

B

A

ED

CGiven: AD ║EC, BD BCProve: ∆ABD ∆EBC

Statements Reasons