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

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Question

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

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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.

Compare your answer with the correct one above

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