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On this page, we hope to clear up problems that you might have with proving right triangles congruent.  Right triangles are special triangles that contain one right angle.  With right triangles, we name the sides of the triangle.  The two sides that include the right angle are called legs and the side opposite the right angle is called the hypotenuse.  Scroll down or click any of the links below to start understanding congruent right triangles better! Leg-Leg Theorem Leg-Acute Angle Theorem Hypotenuse-Acute Angle Theorem Hypotenuse-Leg Postulate Quiz on Congruent Right Triangles The Leg-Leg Theorem is a rule specially designed for use with right triangles.  (If anyone cares, it is actually the Side-Angle-Side rule.)  It states if the legs of one right triangle are congruent to the legs of another right triangle, the two right triangles are congruent. 1. Problem: Prove that triangle ABC is congruent to triangle DEF. Solution: Segment AB and segment DE, which are both legs of their respective triangles, are congruent because that information is given. Segment BC and segment EF, which are both legs of their respective triangles, are congruent because that information is given. The figure also denotes both triangles as right triangles. Because of that fact, and the fact that both legs are congruent, triangle ABC and triangle DEF are congruent by the Leg-Leg Theorem. Back to Top  The Leg-Acute Angle Theorem is a rule specially designed for use with right triangles.  (If anyone cares, it is actually the Angle-Side-Angle rule.)  It states if a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent. 1. Problem: Prove triangle JKL is congruent to triangle MNO. Solution: Segment KL and segment NO, which are legs of their respective triangles, are congruent because that information is given. Angle L is congruent to angle O, which are acute angles of their respective triangles, are congruent because that information is given in the figure. Since there is one leg and one acute angle in each triangle that is congruent to another leg and another acute angle in the other triangle, both of which are right, triangle JKL is congruent to triangle MNO by the Leg-Acute Angle Theorem. Back to Top  The Hypotenuse-Acute Angle Theorem is a rule specially designed for use with right triangles.  (If anyone cares, it is actually the Angle-Angle-Side rule.)  It states if the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, the two triangles are congruent. 1.   Problem: Prove triangle PQR is congruent to triangle STU. Solution: Segment PR and segment SU, which are the hypotenuses of their respective triangles, are congruent because that information is given in the figure. Angle R and angle U, which are both acute angles, are congruent because that information is given in the figure. With the above information, which says the hypotenuses and one of one of the acute angles in each triangle are congruent, you have proved that triangle PQR is congruent to triangle STU by the Hypotenuse-Acute Angle Theorem. Back to Top  The Hypotenuse-Leg Postulate is a rule that you can use with right triangles only.  This rule is considered a postulate because it is not based on any other rules, as the theorems discussed above have been.  It states if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. 1.   Problem: State why, after looking at the figure, the following conclusion has been reached: Triangle QRS is congruent to triangle XYZ. Solution: The hypotenuses of each triangle are con- gruent. Each triangle also has a congruent leg (in this case, RS and YZ are given in the figure as congruent). With each triangle having a congruent hypotenuse and one congruent leg, the two triangles can be shown to be congruent by the Hypotenuse-Leg Postulate. Back to Top  Take the Quiz on congruent right triangles.  (Very useful to review or to see if you've really got this topic down.)  Do it!