Congruence Criteria Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Congruence Criteria.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: You need at most three well-chosen measurements to completely determine a triangle—and to prove two triangles are identical.

Common stuck point: SSA (two sides and a non-included angle) is NOT a valid criterion—it can produce two different triangles (the ambiguous case).

Worked Examples

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.

Solution

  1. 1
    Step 1: List what is known: all three pairs of corresponding sides are equal — AB = DE, BC = EF, AC = DF.
  2. 2
    Step 2: Identify the applicable congruence criterion. When all three sides of one triangle equal the corresponding sides of another, we use SSS (Side-Side-Side).
  3. 3
    Step 3: Conclude: By SSS, \triangle ABC \cong \triangle DEF.

Answer

\triangle ABC \cong \triangle DEF by SSS.
SSS (Side-Side-Side) congruence states that if all three sides of one triangle are equal in length to the corresponding sides of another triangle, the triangles are congruent. This works because the shape of a triangle is completely determined by its three side lengths.

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?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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 2

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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Background Knowledge

These ideas may be useful before you work through the harder examples.

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