Geometry: Congruent Right Triangles

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. Read on or follow 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

Leg-Leg Theorem

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.

Example:

1.  Problem: Prove that triangle ABC in the accompanying figure is
             congruent to triangle DEF.
   Solution: Segment AB and segment DE, which are both
             legs of their respective triangles, are congruent be-
             cause 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 tri-
             angles, and because of that fact, and the fact that
             both legs are congruent, triangle ABC and
             triangle DEF are congruent by the Leg-Leg
             Theorem.

Leg-Acute Angle Theorem

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.

Example:

1.  Problem: Prove triangle JKL in the accompanying figure 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.

Hypotenuse-Acute Angle Theorem

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.

Example:

1.  Problem: Prove triangle PQR in the accompanying figure 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 that hypotenuses
             and 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.

Hypotenuse-Leg Postulate

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.

Example:

1.  Problem: State why, after looking at the accompanying figure,
             the following conclusion has been reached:
             Triangle QRS is congruent to triangle XYZ.
   Solution: The hypotenuses of each triangle are congruent.  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.

Take the quiz on congruent right triangles. The quiz is very useful for either review or to see if you've really got the topic down.

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Math for Morons Like Us -- Geometry: Congruent Right Triangles
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