Geometry

Triangle Congruence Theorems

Congruent Triangles

- Congruent triangles have three congruent sides

and and three congruent angles. - However, triangles can be proved congruent

without showing 3 pairs of congruent sides and

angles.

The Triangle Congruence Postulates Theorems

Theorem

- If two angles in one triangle are congruent to

two angles in another triangle, the third angles

must also be congruent.

- Think about it they have to add up to 180.

A closer look...

- If two triangles have two pairs of angles

congruent, then their third pair of angles is

congruent.

- But do the two triangles have to be congruent?

Example

Why arent these triangles congruent? What do

we call these triangles?

- So, how do we prove that two triangles really are

congruent?

ASA (Angle, Side, Angle)

- If two angles and the included side of one

triangle are congruent to two angles and the

included side of another triangle, . . .

then the 2 triangles are CONGRUENT!

AAS (Angle, Angle, Side) Special case of ASA

- If two angles and a non-included side of one

triangle are congruent to two angles and the

corresponding non-included side of another

triangle, . . .

then the 2 triangles are CONGRUENT!

SAS (Side, Angle, Side)

- If in two triangles, two sides and the included

angle of one are congruent to two sides and the

included angle of the other, . . .

then the 2 triangles are CONGRUENT!

SSS (Side, Side, Side)

- In two triangles, if 3 sides of one are congruent

to three sides of the other, . . .

then the 2 triangles are CONGRUENT!

HL (Hypotenuse, Leg)

- If both hypotenuses and a pair of legs of two

RIGHT triangles are congruent, . . .

then the 2 triangles are CONGRUENT!

HA (Hypotenuse, Angle)

- If both hypotenuses and a pair of acute angles of

two RIGHT triangles are congruent, . . .

then the 2 triangles are CONGRUENT!

LA (Leg, Angle)

- If both hypotenuses and a pair of acute angles of

two RIGHT triangles are congruent, . . .

then the 2 triangles are CONGRUENT!

LL (Leg, Leg)

- If both pair of legs of two RIGHT triangles are

congruent, . . .

then the 2 triangles are CONGRUENT!

Example 1

- Given the markings on the diagram, is the pair of

triangles congruent by one of the congruency

theorems in this lesson?

D

E

F

Example 2

- Given the markings on the diagram, is the pair of

triangles congruent by one of the congruency

theorems in this lesson?

Example 3

- Given the markings on the diagram, is the pair of

triangles congruent by one of the congruency

theorems in this lesson?

Example 4

- Why are the two triangles congruent?
- What are the corresponding vertices?

SAS

?A ? ? D

?C ? ? E

?B ? ? F

Example 5

A

- Why are the two triangles congruent?
- What are the corresponding vertices?

SSS

B

D

?A ? ? C

?ADB ? ? CDB

C

?ABD ? ? CBD

Example 6

- Given

Are the triangles congruent?

Why?

S S S

Example 7

- Given

m?QSR m?PRS 90

- Are the Triangles Congruent?

Why?

R H S

?QSR ? ?PRS 90

Summary

ASA - Pairs of congruent sides contained between

two congruent angles

AAS Pairs of congruent angles and the side not

contained between them.

SAS - Pairs of congruent angles contained between

two congruent sides

SSS - Three pairs of congruent sides

Summary --- for Right Triangles Only

HL Pair of sides including the Hypotenuse and

one Leg HA Pair of hypotenuses and one acute

angle LL Both pair of legs LA One pair of

legs and one pair of acute angles

THE END!!!