Triangle congruence

1. What is Triangle Congruence?

Triangle congruence means that two triangles are exactly the same in shape and size. If you place one congruent triangle on top of another, they would perfectly match up. This implies that all corresponding sides and all corresponding angles are equal. We use the symbol $ \cong $ to denote congruence. For example, if triangle ABC is congruent to triangle DEF, we write $ \triangle ABC \cong \triangle DEF $ .

2. Congruence Postulates and Theorems

To prove that two triangles are congruent, we don't always need to show that all three sides and all three angles are equal. There are several postulates and theorems that allow us to prove congruence with fewer pieces of information.

2.1 Side-Side-Side (SSS) Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Condition: All three pairs of corresponding sides are equal in length.
Example: If side AB = side DE, side BC = side EF, and side CA = side FD, then $ \triangle ABC \cong \triangle DEF $ .

2.2 Side-Angle-Side (SAS) Postulate

If two sides and the included angle (the angle between those two sides) of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Condition: Two pairs of corresponding sides and the angle between them are equal.
Important: The angle must be included (between the two sides).
Example: If side AB = side DE, angle B = angle E, and side BC = side EF, then $ \triangle ABC \cong \triangle DEF $ .

2.3 Angle-Side-Angle (ASA) Postulate

If two angles and the included side (the side between those two angles) of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Condition: Two pairs of corresponding angles and the side between them are equal.
Important: The side must be included (between the two angles).
Example: If angle A = angle D, side AB = side DE, and angle B = angle E, then $ \triangle ABC \cong \triangle DEF $ .

2.4 Angle-Angle-Side (AAS) Theorem

If two angles and a non-included side (a side not between those two angles) of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

Condition: Two pairs of corresponding angles and a side not between them are equal.
Note: This is often considered a theorem because if you know two angles, the third angle is automatically determined ( $180^\circ - \text{angle } 1 - \text{angle } 2$ ). So, AAS can be thought of as an extension of ASA.
Example: If angle A = angle D, angle B = angle E, and side BC = side EF (where BC and EF are not between angles A/B and D/E respectively), then $ \triangle ABC \cong \triangle DEF $ .

2.5 Hypotenuse-Leg (HL) Theorem

This theorem is specific to right-angled triangles. If the hypotenuse and one leg of a right-angled triangle are congruent to the hypotenuse and one leg of another right-angled triangle, then the triangles are congruent.

Condition: Applicable only to right triangles, where the hypotenuses are equal, and one pair of corresponding legs are equal.
Example: In right triangles ABC and DEF (with right angles at B and E respectively), if hypotenuse AC = hypotenuse DF, and leg BC = leg EF, then $ \triangle ABC \cong \triangle DEF $ .

3. What Does NOT Prove Congruence?

Angle-Angle-Angle (AAA): Simply having all three angles equal only proves that triangles are similar (same shape, different size), not necessarily congruent. For example, an equilateral triangle with side length 5 and an equilateral triangle with side length 10 both have three $60^\circ$ angles, but are not congruent.
Side-Side-Angle (SSA): This case (also sometimes called ASS) does not guarantee congruence. If the angle is not included between the two sides, it can lead to two possible different triangles, making it ambiguous. There is an exception with HL for right triangles because the right angle provides a fixed relationship.