Using Congruence in Proofs: Practice

CPCTC describes corresponding parts after triangle congruence is known.

For example, from

it gives

A criterion establishes the whole-triangle match first, for example

CPCTC then supplies the matching sides or angles needed for the goal.

Match $B$ to $Q$, $C$ to $P$, and $A$ to $R$.

The three-letter angle name maps as

The same length equals itself:

This reflexive equality is available before either triangle has been proved congruent to the other.

Each angle lies between the two named sides.

The correspondence is

so the two side pairs and included angle establish SAS.

The desired equality cannot supply its own justification.

A valid argument first establishes triangle congruence from other known data. CPCTC then gives the target equality:

An auxiliary segment can create useful triangles, such as a diagonal $PR$.

It supplies a shared part

but other equality claims still need reasons.

An angle is determined by its two rays, not just its vertex.

If ray $AB$ is ray $AD$ and ray $AC$ is ray $AE$, then

The compass settings establish three side pairs, such as

SSS applies without assuming the target angle already equals the source angle.

The source and target angles are corresponding parts of the established congruent triangles.

In the order

the conclusion is

The shared baseline lies between the two known angle vertices.

It is the included side, with equality

The angle-side-angle data give ASA.

Hypotenuse-Leg requires two right triangles.

The side opposite each right angle is its hypotenuse, and the required lengths match, for example