Question

Give me a quick refresher on congruence in geometry

Answer 1

Congruence in geometry refers to the idea that two shapes or figures are congruent if they have the same size and shape. This means that all corresponding sides and angles of the two figures are equal. For instance, two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in measure. This is often denoted as ΔABC ≅ ΔDEF, where Δ is the symbol for a triangle and ABC and DEF are the vertices of the two triangles.There are several postulates and theorems that help determine if two shapes are congruent:

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.

2. Side-Angle-Side (SAS) Postulate: 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.

3. Angle-Side-Angle (ASA) Postulate: 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.

4. Angle-Angle-Side (AAS) Theorem: If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.

5. Hypotenuse-Leg (HL) Theorem: Applicable only to right triangles. If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

Remember, congruence is about size and shape, but not position or orientation. So a figure could be rotated, reflected or translated and still be congruent to the original.