Triangle Congruence Postulates: SAS, ASA, SSS, AAS, HL

Congruent triangles are triangles with identical sides and angles. The three sides of one are exactly equal in measure to the three sides of another. The three angles of one are each the same angle as the other.

Table Of Contents

1. Triangle Congruence Postulates
2. Included Parts
3. Side Side Side Postulate
4. Side Angle Side Postulate
5. Angle Side Angle Postulate
6. Angle Angle Side Theorem
7. HL Postulate
8. Proof Using Congruence

Triangle Congruence Postulates

Five ways are available for finding two triangles congruent:

1. SSS, or Side Side Side
2. SAS, or Side Angle Side
3. ASA, or Angle Side Side
4. AAS, or Angle Angle Side
5. HL, or Hypotenuse Leg, for right triangles only

Included Parts

An included angle lies between two named sides. In △CAT below, included ∠A is between sides t and c:

An included side lies between two named angles of the triangle.

Side Side Side Postulate

A postulate is a statement taken to be true without proof. The SSS Postulate tells us,

Congruence of sides is shown with little hatch marks, like this: ∥. For two triangles, sides may be marked with one, two, and three hatch marks.

If △ACE has sides identical in measure to the three sides of △HUM, then the two triangles are congruent by SSS:

Side Angle Side Postulate

The SAS Postulate tells us,

△HUG and △LAB each have one angle measuring exactly 63°. Corresponding sides g and b are congruent. Sides h and l are congruent.

A side, an included angle, and a side on △HUG and on △LAB are congruent. So, by SAS, the two triangles are congruent.

Angle Side Angle Postulate

This postulate says,

We have △MAC and △CHZ, with side m congruent to side c. ∠A is congruent to ∠H, while ∠C is congruent to ∠Z. By the ASA Postulate these two triangles are congruent.

Angle Angle Side Theorem

We are given two angles and the non-included side, the side opposite one of the angles. The Angle Angle Side Theorem says,

Here are congruent △POT and △LID, with two measured angles of 56° and 52°, and a non-included side of 13 centimeters:

[construct as described]

By the AAS Theorem, these two triangles are congruent.

HL Postulate

Exclusively for right triangles, the HL Postulate tells us,

The hypotenuse of a right triangle is the longest side. The other two sides are legs. Either leg can be congruent between the two triangles.

Here are right triangles △COW and △PIG, with hypotenuses of sides w and i congruent. Legs o and g are also congruent:

[insert congruent right triangles left-facing △COW and right facing △PIG]

So, by the HL Postulate, these two triangles are congruent, even if they are facing in different directions.

Proof Using Congruence

Given: △MAG and △ICG

MC ≅ AI

AG ≅ GI

Prove: △MAG ≅ △ICG

Statement Reason

MC ≅ AI Given

AG ≅ GI

∠MGA ≅ ∠ IGC Vertical Angles are Congruent

△MAG ≅ △ICG Side Angle Side

If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Next Lesson:

Triangle Congruence Theorems