Hypotenuse Leg Theorem

Proving Congruent Right Triangles

The hypotenuse leg theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles.

Examples

Example 1

Given:AB = XZ, CB = XY, ACB = ZYX = 90°

Prove: ABC XYZ

  • ABC and XZY are right triangles since they both have a right angle
  • AB = XZ (hypotenuse)  reason: given
  • AC = ZY (leg) reason: given
  • ABC XYZ by the hypotenuse leg theorem which states that two right triangles are congruent if their hypotenuses are congruent and a corresponding leg is congruent.
Hypotenuse Leg Theorem
Example 2

The following proof simply shows that it does not matter which of the two (corresponding) legs in the two right triangles are congruent

  • ABC and XZY are right triangles since they both have a right angle
  • AB = XZ (hypotenuse)  reason: given
  • CB = XY (leg) reason: given
  • ABC XYZ by the hypotenuse leg theorem which states that two right triangles are congruent if their hypotenuses are congruent and a corresponding leg is congruent.
Hypotenuse Leg Theorem Example