PSAT Math - How to find if acute / obtuse triangles are congruent

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

I) $\Delta MNO$ and $\Delta PQR$ have the same perimeter

II) $m \angle M + m \angle O = m \angle P + m \angle R$

III) $m \angle M + m \angle N = m \angle P + m \angle Q$

Statement III only

Statement II only

Statement I only

Any of the three statements is enough to prove congruence.

None of these statements is enough to prove congruence.

Explanation

If $\Delta MNO$ and $\Delta PQR$ have the same perimeter, , and , it follows that . The three triangles have the same sidelengths, setting the conditions for the Side-Side-Side Congruence Postulate.

If $m \angle M + m \angle O = m \angle P + m \angle R$ , then, since the sum of the degree measures of both triangles is the same (180 degrees), it follows that $m \angle N = m \angle Q$ . Since $\angle N$ and $\angle Q$ are congruent included angles of congruent sides, this sets the conditions for the SAS Congruence Postulate.

In both of the above cases, it follows that $\Delta MNO \cong \Delta PQR$ .

However, similarly to the previous situation, if $m \angle M + m \angle N = m \angle P + m \angle Q$ , then it follows that $m \angle O = m \ m \angle R$ , meaning that we have congruent sides and congruent nonincluded angles. However, this is not sufficient to prove congruence.

"Statement III" is the correct response.

How to find if acute / obtuse triangles are congruent

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