# 2.4 – Congruence Postulates

## Objectives

• Discover the congruence shortcuts (SSS, SAS, and ASA postulates).
• Use the AAS theorem to determine whether two triangles are congruent.
• Experiment with SSA and AAA and discover why they do not work as congruence shortcuts.

## Key Terms

• Included Angle – An angle between two sides of a triangle
• AAS Theorem – AAS stands for “angle-angle-side”
• A theorem stating that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another, then the triangles are congruent.
• This is a THEOREM because we can prove it using the “Third Angle Theorem”
• ASA Postulate – ASA stands for “angle-side-angle”
• A postulate stating that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
• SAS Postulate – SAS stands for “side-angle-side”
• A postulate stating that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
• SSS Postulate – SSS stands for “side-side-side”
• A postulate stating that if the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
• Triangles with three pairs of congruent sides will always be congruent.
• You can prove the SSS Postulate by using the other congruence postulates; so, it can be called a Theorem.
• Triangle Congruence Postulate – A statement that proves triangles are congruent without requiring that the measures of all six pairs of corresponding parts be known.
• Third Angle Theorem – If two angles of one triangle are congruent to two angles of another triangle, the third angles are also congruent.
• You can combine ASA and AAS to make one general rule:  If two angles and any side of one triangle are congruent to the corresponding two angles and side of another triangle, the two triangles are congruent.

## Notes

Review
• What’s the difference between a postulate and a theorem?
• A postulate (axiom) is a statement that is assumed to be true without proof.
• A theorem is a statement that has been proven.
Congruence
• If it is possible, it could always be true.
• If it’s not possible, it’s not always true.

• To determine congruence, you trace a triangle clockwise or counter-clockwise, looking for sides and angles.
• You can only skip ONE, either a side OR an angle, when you trace the triangle in either direction.
• SSA (Side-Side-Angle) or A-S-S does NOT guarantee (or prove) congruence!
• Remember: ASS means donkey, not congruence!
• AAA (Angle-Angle-Angle) does NOT guarantee (or prove) congruence!
• Remember AAA tows your car, not your triangles!
• See below (the triangles have the same angle measures, but one is larger than the other (not congruent)!

• Proof by Contradiction (Indirect Proof): Side-Side-Side
• We want to prove: If three pairs of segments are congruent, they always make two congruent triangles.
• So we are proving: If three pairs of segments are congruent, it is impossible to make two triangles that are not congruent.
• Proof by Contradiction (Indirect Proof): Side-Angle-Side
• We want to prove: If two pairs of segments and their included angles are congruent, they always make two congruent triangles.
• So we are proving: If two pairs of segments and their included angles are congruent, it is impossible to make two triangles that are not congruent.
• Proof by Contradiction (Indirect Proof): Angle-Side-Angle
• We want to prove: If two pairs of angles and their included sides are congruent, they always make two congruent triangles.
• So we are proving: If two pairs of angles and their included sides are congruent, it is impossible to make two triangles that are not congruent.

Examples
• Example of ASA

Answer: $\overline{AB} \cong \overline{XY}$

• Example of SAS

$90^{\circ} (angle)-40^{\circ} (angle)-12 (side)$