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Congruent Triangles - Hypotenuse and leg of a right triangle. (HL)

Definition: Two right triangles are congruent if the hypotenuse and one corresponding leg are equal in both triangles.

There are five ways to test that two triangles are congruent. This is one of them (HL). For a list see Congruent Triangles. If, in two right triangles the hypotenuse and one leg are equal, then the triangles are congruent.

Try this Drag any orange dot at P,Q,R. The other triangle LMN will change to remain congruent to the triangle PQR.

Notice that the the hypotenuse and leg are drawn in thick blue lines to indicate they are the elements being used to test for congruence.

Notice that, since we know the hypotenuse and one other side, the third side is determined, due to Pythagoras' Theorem. So this is really a version of the SSS case. (side-side-side).

What does this mean?

Because the triangles are congruent:
  1. The remaining third sides are equal (PQ=LM)
  2. The other two angles are equal (Q=M and R=N)

But don't forget:

Two triangles, congruent because all three sides are the same length, but one flipped and rotated Congruent triangles can be rotated and/or mirror images of each other (reflected). (See Congruent triangles.)

In the figure on the right, the two triangles have all three corresponding sides equal in length and so are still congruent, even though one is the mirror image of the other and rotated.

Other congruence topics

Congruent Triangles

Congruent Polygons