Congruent Triangles and CPCTC

Once you can prove two triangles congruent, a whole second conclusion comes for free. Congruent triangles are identical copies, so every pair of matching parts — all three sides and all three angles — must be congruent. That final move has a name you will use constantly: CPCTC, "Corresponding Parts of Congruent Triangles are Congruent."

The proof always has the same shape. First, gather enough matching parts to invoke one of the congruence shortcuts:

SSS — three pairs of sides.
SAS — two pairs of sides and the pair of angles between them.
ASA — two pairs of angles and the pair of sides between them.

Then state the triangles congruent by that criterion, and finish by naming the one extra pair you were asked to prove — justified by CPCTC.

Reading the marks

In the figure, congruent parts wear matching marks: equal sides carry the same number of tick marks, equal angles the same number of arcs, and a small square sits in any right angle. Those marks are the givens — they tell you exactly which parts you may use, and therefore which criterion fits. The part you are asked to prove is left unmarked; it is the one CPCTC delivers at the end.

Build the proof

Read the Given and Prove, then build the argument one line at a time. Work in an order that respects the logic: you cannot claim the triangles congruent until their parts are established, and you cannot use CPCTC until the triangles are congruent. Click a vertex in the figure to find where it appears in the proof, or click a statement to see which parts of the figure it is about. Only a step that genuinely follows will lock into place.

Given: $\overline{AB} \cong \overline{DE}$ , $\overline{BC} \cong \overline{EF}$ , and $\overline{CA} \cong \overline{FD}$

Prove: $\angle C \cong \angle F$

Statements	Reasons
1.
2.
3.
4.
5.

Row 1: choose the reason that justifies this statement.

Reasons

Placed 0 of 5 steps.

Two triangles with their given congruent parts marked (matching ticks on congruent sides, matching arcs on congruent angles, a square at a right angle). Build a two-column proof that

\angle C \cong \angle F

: each statement is given in order and you supply the reason that justifies it. Each row is checked against the logic of the proof, so only a step that genuinely follows will seat.

Key result

To prove a single pair of parts congruent, prove the two triangles congruent first. Collect the matching parts the marks give you, pick the criterion they fit — SSS, SAS, or ASA — state the congruence, then invoke CPCTC to conclude the remaining pair. The order matters: parts, then triangles, then the corresponding part.

For teachers

Standard, common misconceptions, and suggested use will live here.