Practice Congruence Criteria in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

Five sets of conditions that guarantee two triangles are congruent: SSS (three pairs of equal sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse-leg for right triangles).

Imagine building a triangle from sticks and hinges. If you fix all three side lengths (SSS), there's only one triangle you can make. If you fix two sides and the angle between them (SAS), the triangle is locked in place. You don't need all six measurements—just the right three.

Example 1

easy
Two triangles share the following information: AB = DE = 5, BC = EF = 7, AC = DF = 9. Are the triangles congruent? State the congruence criterion used.

Example 2

medium
In right triangles \triangle PQR and \triangle XYZ, both have a right angle. The hypotenuse PR = XZ = 13 and leg PQ = XY = 5. Are the triangles congruent? Which criterion applies?

Example 3

easy
Match each situation to the correct congruence criterion (SSS, SAS, ASA, AAS, HL): Two triangles have two angles and the included side equal.

Example 4

hard
Explain why SSA (two sides and a non-included angle) is NOT a valid congruence criterion by giving a counterexample or geometric explanation.
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