PSAT Math : How to find if right triangles are congruent

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #82 : Right Triangles

You are given triangles   and ,with  and  both right angles, and . Which of these statements, along with what you are given, is not enough to prove that ?

I) 

II) 

III)  and  have the same area.

Possible Answers:

Any of the three statements is enough to prove congruence.

None of the three statements is enough to prove congruence.

Statement I only

Statement III only

Statement II only

Correct answer:

Any of the three statements is enough to prove congruence.

Explanation:

, and the right angles are  and , so we have two right triangles with congruent legs.

If we also know that , then the hypotenuses of the right triangles are also congruent, and this sets up the conditions of the Hypotenuse-Leg Theorem.

If we also know that , then, along with the fact that  (both being right angles) and nonincluded sides , the conditions of the Angle-Angle-Side Theorem are set up.

If we also know  and  have the same area, we can demonstrate that the other legs are congruent. The area of a right triangle is half the product of its legs, and since we have the same areas,

Since ,

The legs and the included angles (the right angles) are congruent, thus setting up the conditions for the Angle-Side-Angle Postulate. 

In all three cases, congruence follows, so the correct response is "Any of the three statements is enough to prove congruence."

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