Congruent triangles problems look like “proofs,” but most of the time they’re a practical shortcut: once you show two triangles match exactly, you get a bunch of equal sides and angles for free. The workflow is: prove congruence → use CPCTC → solve for x.
Not sure which congruence test applies?
Try our ai math solver to identify the right test (SSS/SAS/ASA/AAS/HL), write the congruence statement, and finish with CPCTC.
Two triangles are congruent if they have the same shape and the same size. That means all corresponding sides are equal and all corresponding angles are equal.
Key payoff: Once △ABC ≅ △DEF, then AB = DE, BC = EF, AC = DF and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.
Side-Side-Side: all three sides match.
Side-Angle-Side: two sides and the included angle match.
Angle-Side-Angle: two angles and the included side match.
Angle-Angle-Side: two angles and a non-included side match.
Hypotenuse-Leg: in right triangles, if the hypotenuse and one leg match, the triangles are congruent.
Notice what is not on the list: SSA and AAA. SSA is ambiguous (can make two different triangles), and AAA only proves similarity (same shape) but not size.
The congruence statement is not decoration—it encodes correspondence. If you write △ABC ≅ △DEF, you are claiming: A ↔ D, B ↔ E, C ↔ F. Every CPCTC equation must follow that matching.
Problem: In triangles △ABC and △DEF, AB = DE, ∠B = ∠E, and BC = EF. Side AB is (2x + 3) and DE is (5x − 9). Find x.
The givens are two sides with the included angle: SAS proves congruence. Since AB corresponds to DE, set them equal:
Check: AB = 2(4)+3 = 11 and DE = 5(4)−9 = 11.
Problem: Two right triangles have hypotenuse 13 and one leg 5. Show the triangles are congruent and find the other leg.
HL proves congruence (right triangle + hypotenuse + leg). To find the other leg, use Pythagorean theorem:
In many congruent triangle setups, HL is the quickest path because right angles give you a built-in angle match.
CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. In plain language: once the triangles are proved congruent, you can equate any matching sides and angles.
How CPCTC is used in algebra problems:
When a diagram feels busy, you don’t need more steps—you need the right starting point. Use this checklist to decide quickly which congruence test is available.
If your givens look like SSA, pause and verify whether the triangles are right triangles. Otherwise, SSA can create two different triangles, so congruence is not guaranteed.
Problem: In triangles △ABC and △DEF, ∠A = ∠D, ∠B = ∠E, and AB = DE. Prove the triangles are congruent and find ∠C if ∠F = 52°.
Two angles and the included side are given, so this is ASA. Once △ABC ≅ △DEF, corresponding angles are equal. Vertex C corresponds to vertex F, so ∠C = ∠F.
This is a classic CPCTC usage: prove congruence first, then “transfer” an angle measurement from one triangle to the other.
Problem: Suppose △PQR ≅ △XYZ. If PQ = (x + 7), XY = 16, and ∠R = (2y + 10)° while ∠Z = 64°, solve for x and y.
From the congruence statement, P ↔ X, Q ↔ Y, R ↔ Z. That means PQ corresponds to XY and ∠R corresponds to ∠Z. Set up two equations:
Notice how the congruence statement is doing the heavy lifting: it tells you exactly which parts are allowed to be set equal.
SSA is ambiguous. Unless it is a right triangle with the hypotenuse known (HL), SSA does not guarantee congruence.
Writing △ABC ≅ △DFE and then using AB = DE is a mismatch. Your congruence statement controls every pairing.
The angle in SAS/ASA must sit between the two sides (or angles) you are using. Otherwise the test may not apply.
Start with the easiest pair you can justify. Once you get one test, CPCTC gives you the rest.
If you want a concise summary of the congruence criteria (and why SSA fails), the reference overview on triangle congruencecan be helpful when you just need to confirm which test applies.
Paste the givens and get the congruence proof outline + the CPCTC equation for x.
Try AI Math Solver →