Congruent Triangles have six sets of corresponding parts!

Three sets of corresponding sidesThree sets of corresponding angles

You don’t need to know all six sets of corresponding parts to determine congruence.

So, what information about two triangles is enough to prove

congruence?

First: Some Terminology!

• Included Angle – The angle located BETWEEN two adjacent sides

Angle B is the included angleof sides AB and BC

• Included Side – The side located BETWEEN two angles

Side MN is the included side between Angle M and Angle N

Can you make two triangles that are NOT congruent using just

the given information?

If you can make two non-congruent triangles, you don’t have enough information to prove congruence

If you can’t make two non-congruent triangles, then the information MUST BE ENOUGH TO PROVE CONGRUENCE!

• Triangle ABC and DEF both have side lengths of 5cm, 8cm and 11cm.

SSS Triangle Postulate

• Side – Side – Side (SSS) IS ENOUGH TO PROVE CONGRUENCE.

These three measurements “LOCK IN” the the angle measurements so all corresponding

parts are congruent.

Triangle ABC and Triangle DEF both have angles that measure 40o, 60o and 80o

• Angle – Angle – Angle (AAA) IS NOT ENOUGH TO PROVE CONGRUENCE.

• It is enough to prove similarity of triangles

• A dilation keeps the angles congruent but the side lengths would change by the scale factor.

• Triangle ABC and DEF both have two sides measuring 5cm and 11 cm with an included angle of 60o.

SAS Triangle Postulate

• Side – Angle – Side (SAS) IS ENOUGH TO PROVE CONGRUENCE.

These three measurements “LOCK IN” the other three measurements so all corresponding

parts are congruent.

• Triangle ABC and Triangle DEF both have angles that measure 40o and 60o and an included side of 11 cm.

ASA Triangle Postulate

• Angle – Side – Angle (ASA) IS ENOUGH TO PROVE CONGRUENCE.

These three measurements “LOCK IN” the other three measurements so all corresponding parts are congruent.

• Triangles ABC and DEF both have two angles that are 60o and 80o and a non-included side of 8 cm.

AAS Triangle Postulate

• Angle – Angle – Side (AAS) IS ENOUGH TO PROVE CONGRUENCE.

These three measurements “LOCK IN” the other three measurements so all corresponding parts are congruent.

• Triangles ABC and DEF both have two sides that are 8cm and 11cm and a non-included angle of 40o.

There is a “Swing Effect” forthat 2nd side!

40o 40o

TWO non-congruent triangles can both have these two sides and

the non-included angle.

• Side – Side – Angle (SSA) IS NOT ENOUGH TO PROVE CONGRUENCE.

These three measurements do not lock in the other three measurements because of that “swing effect.”

Therefore, we cannot say two triangles are congruent based on SSA.

• Right Triangle ABC and Right Triangle DEF both have a leg 6cm and a hypotenuse 12cm.

12 cm

6cm

• Hypotenuse-Leg (H-L) IS ENOUGH TO PROVE CONGRUENCE for

RIGHT TRIANGLES.

The Pythagorean Theorem “LOCKS IN” the third side because a2+b2=c2 is true for all right triangles. HL is actually a special case of the SSS Postulate.

H-L TRIANGLE POSTULATE

SUMMARYTRIANGLE CONGRUENCE POSTULATES

Not Enough Information to Prove Triangle Congruence

SSS : Side-Side-Side AAA – proves similarity – the angles make the triangle shape but do not determine the size of the triangle

SAS: Side-Angle-Side SSA – could be two different triangles due to a “swing effect”

ASA: Angle-Side-Angle

AAS: Angle-Angle-Side

HL: Hypotenuse-Leg (special case of SSS)

Are the two triangles congruent?If so, which postulate is shown?