Understanding Congruence of Triangles: Definitions and Rules

Q: What does CPCT stand for?

<div class="fusion-layout-column fusion_builder_column fusion-builder-column-39 fusion_builder_column_1_1 1_1 fusion-flex-column" style="--awb-bg-size:cover;--awb-width-large:100%;--awb-margin-top-large:0px;--awb-spacing-right-large:1.92%;--awb-margin-bottom-large:20px;--awb-spacing-left-large:1.92%;--awb-width-medium:100%;--awb-order-medium:0;--awb-spacing-right-medium:1.92%;--awb-spacing-left-medium:1.92%;--awb-width-small:100%;--awb-order-small:0;--awb-spacing-right-small:1.92%;--awb-spacing-left-small:1.92%;"> CPCT stands for “Corresponding Parts of Congruent Triangles”

Two triangles are congruent if their corresponding sides and angles are equal. They can be flipped, rotated, or moved, but they will still look the same. The symbol for congruence is ‘ $\boldsymbol{\cong}$ ‘.

In mathematics, congruence means two shapes are the same in both size and shape. We can tell if two triangles are congruent by following four main rules, and we only need to know three out of six measurements to prove it. The matching sides and angles of congruent triangles are always equal.

Congruence in Math

Congruence in math refers to shapes that can be moved or flipped to fit exactly on top of each other. Two shapes are congruent if they have the same shape and size, like a shape and its mirror image.

Congruent Triangles

A triangle is a three-sided polygon. Two triangles are congruent if their sides are the same length and their angles are the same measure. For example, in the figure, triangles Δ ABC and Δ PQR are congruent if:

Vertices match: A with P, B with Q, C with R
Sides match: AB = PQ, BC = QR, AC = PR
Angles match: ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R

Congruent triangles have the same area and perimeter.

CPCT Full Form

CPCT stands for “Corresponding Parts of Congruent Triangles.” This means that the matching parts of congruent triangles are equal. When solving problems about triangles, we use CPCT to quickly refer to this concept.

CPCT Rules in Math

After proving triangles congruent, we can predict the other dimensions without measuring. Here are the rules:

SSS (Side-Side-Side): If all three sides of one triangle match the three sides of another, they are congruent.

SAS (Side-Angle-Side): If two sides and the included angle of one triangle match the corresponding parts of another, they are congruent.

ASA (Angle-Side-Angle): If two angles and the included side of one triangle match the corresponding parts of another, they are congruent.

AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle match the corresponding parts of another, they are congruent.

RHS (Right angle-Hypotenuse-Side): If the hypotenuse and one side of a right-angled triangle match the corresponding parts of another, they are congruent.