Geometry

Triangle Congruence Theorems

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.

Congruent Triangles

The Triangle Congruence

Postulates &Theorems

LA HA LL HL

FOR RIGHT TRIANGLES ONLY

AAS ASA SAS SSS

FOR ALL TRIANGLES

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?

85° 30°

85° 30°

Example

30°

30°

Why aren’t 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!

F

E

D

A

C

B

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!

F

E

D

A

C

B

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!

F

E

D

A

C

B

SSS (Side, Side, Side)

In two triangles, if 3 sides of one are congruent to three sides of the other, . . .

F

E

D

A

C

B

then

the 2 triangles are

CONGRUENT!

HL (Hypotenuse, Leg)

If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . .

A

C

B

F

E

D

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!

F

E

D

A

C

B

LA (Leg, Angle)

If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .

then

the 2 triangles are

CONGRUENT!

A

C

B

F

E

D

LL (Leg, Leg)

If both pair of legs of two RIGHT triangles are congruent, . . .

then

the 2 triangles are

CONGRUENT!

A

C

B

F

E

D

Example 1

Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

F

E

D

A

C

B

Example 2

Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

A

C

B

F

E

D

Example 3

Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

D

A

C

B

Example 4

Why are the two triangles congruent?

What are the corresponding vertices?

A

B

C

D

E

F SAS

A D

C E

B F

Example 5

Why are the two triangles congruent?

What are the corresponding vertices?

A

B

C

D SSS

A C

ADB CDB

ABD CBD

Example 6

Given:

B C

D A

CDAB

ADBC

CDAB

DABC

CAAC

Are the triangles congruent?

S

S

S

Why?

Example 7

Given: QR PS

R

H

S RSSR

Are the Triangles Congruent?

QSR PRS = 90°

Q

R S

P

T

mQSR = mPRS = 90°

PS QR

Why?

Summary:

ASA - Pairs of congruent sides contained between two congruent angles

SAS - Pairs of congruent angles contained between two congruent sides

SSS - Three pairs of congruent sides

AAS – Pairs of congruent angles and the side not contained between them.

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!!!