Congruence Criteria Math Example 2

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Example 2

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

Solution

  1. 1
    Step 1: Both triangles are right triangles (have a 90° angle). Their hypotenuses are equal: PR=XZ=13PR = XZ = 13. One pair of legs is equal: PQ=XY=5PQ = XY = 5.
  2. 2
    Step 2: Identify the criterion. For right triangles, HL (Hypotenuse-Leg) states: if the hypotenuse and one leg of one right triangle equal those of another right triangle, the triangles are congruent.
  3. 3
    Step 3: Verify HL applies: right angle + equal hypotenuses + equal one leg → HL criterion is satisfied.
  4. 4
    Step 4: Conclude: △PQR≅△XYZ\triangle PQR \cong \triangle XYZ by HL.

Answer

△PQR≅△XYZ\triangle PQR \cong \triangle XYZ by HL (Hypotenuse-Leg).
HL is a special congruence criterion that applies only to right triangles. Once you know the right angle and two sides (hypotenuse and one leg), the third side is determined by the Pythagorean theorem, making the triangle completely determined — and thus congruent to any other right triangle with the same hypotenuse and leg.

About Congruence Criteria

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).

Learn more about Congruence Criteria →

More Congruence Criteria Examples

Example 1 easy

Two triangles share the following information: [formula], [formula], [formula]. Are the triangles co

Example 3 easy

Match each situation to the correct congruence criterion (SSS, SAS, ASA, AAS, HL): Two triangles hav

Example 4 hard

Explain why SSA (two sides and a non-included angle) is NOT a valid congruence criterion by giving a

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