How to find if right triangles are congruent

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PSAT Math › How to find if right triangles are congruent

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1

You are given triangles $\Delta MNO$ and $\Delta PQR$ ,with $\angle N$ and $\angle Q$ both right angles, and . Which of these statements, along with what you are given, is not enough to prove that $\Delta MNO \cong \Delta PQR$ ?

I)

II) $\angle O \cong \angle R$

III) $\Delta MNO$ and $\Delta PQR$ have the same area.

Any of the three statements is enough to prove congruence.

Statement III only

Statement II only

Statement I only

None of the three statements is enough to prove congruence.

Explanation

, and the right angles are $\angle N$ and $\angle Q$ , 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 $\angle O \cong \angle R$ , then, along with the fact that $\angle N \cong \angle Q$ (both being right angles) and nonincluded sides , the conditions of the Angle-Angle-Side Theorem are set up.

If we also know $\Delta MNO$ and $\Delta PQR$ 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,

$\frac{1}{2} (MN)(NO) = \frac{1}{2} (PQ)(QR)$

$2 \times \frac{1}{2} (MN)(NO) =2 \times \frac{1}{2} (PQ)(QR)$

Since ,

$\frac{(MN)(NO)}{MN} = \frac{(PQ)(QR)}{PQ}$

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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