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Example Question #1 : How To Find If Right Triangles Are Congruent
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.
None of the three statements is enough to prove congruence.
Any of the three statements is enough to prove congruence.
Statement II only
Statement I only
Statement III only
Any of the three statements is enough to prove congruence.
, 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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