Prove that in quadrilateral MNOP, if MN = OP and MP = NO, then angle PMN = angle NOP.

March 30, 2026Mathematics

ScanSolve AI Answer

3 stepsMarch 30, 2026

Answer

Proven that \angle PMN = \angle NOP

Step 1: Understand the given information. We are given a quadrilateral $MNOP$ with the following properties: • $\overline{MN} = \overline{OP}$ • $\overline{MP} = \overline{NO}$

We need to prove that $\angle PMN = \angle NOP$ .

Step 2: Draw a diagonal to form two triangles. Draw the diagonal $\overline{NP}$ in the quadrilateral $MNOP$ . This divides the quadrilateral into two triangles: $\triangle MNP$ and $\triangle PON$ .

Step 3: Prove the congruence of $\triangle MNP$ and $\triangle PON$ . Consider $\triangle MNP$ and $\triangle PON$ :

$MN = OP$ (Given)
$MP = NO$ (Given)
$NP = PN$ (Common side to both triangles)

By the Side-Side-Side (SSS) congruence criterion, $\triangle MNP \cong \triangle PON$ .

Step 4: Conclude the equality of corresponding angles. Since $\triangle MNP \cong \triangle PON$ , their corresponding angles are equal. The angle $\angle PMN$ in $\triangle MNP$ corresponds to the angle $\angle NOP$ in $\triangle PON$ . Therefore, $\angle PMN = \angle NOP$ .

This completes the proof.

$\boxed{Proven that \angle PMN = \angle NOP}$

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Understand the given information. We are given a quadrilateral MNOP with the following properties: • MN = OP • MP = NO We need to prove that PMN = NOP.

Asked on March 30, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on March 30, 2026

Step 1: Understand the given information. We are given a quadrilateral MNOP with the following properties: • MN = OP • MP = NO We need to prove that PMN = NOP. Step 2: Draw a diagonal to form two triangles. Draw the diagonal NP in the quadrilateral MNOP. This divides the quadrilateral into two triangles: MNP and PON. Step 3: Prove the congruence of MNP and PON. Consider MNP and PON: 1. MN = OP (Given) 2. MP = NO (Given) 3. NP = PN (Common side to both triangles) By the Side-Side-Side (SSS) congruence criterion, MNP PON. Step 4: Conclude the equality of corresponding angles. Since MNP PON, their corresponding angles are equal. The angle PMN in MNP corresponds to the angle NOP in PON. Therefore, PMN = NOP. This completes the proof. Proven that PMN = NOP

Asked on March 30, 2026Mathematics

ScanSolve AI AnswerAnswered on March 30, 2026

Step 1: Understand the given information. We are given a quadrilateral MNOP with the following properties: • MN = OP • MP = NO We need to prove that PMN = NOP. Step 2: Draw a diagonal to form two triangles. Draw the diagonal NP in the quadrilateral MNOP. This divides the quadrilateral into two triangles: MNP and PON. Step 3: Prove the congruence of MNP and PON. Consider MNP and PON: 1. MN = OP (Given) 2. MP = NO (Given) 3. NP = PN (Common side to both triangles) By the Side-Side-Side (SSS) congruence criterion, MNP PON. Step 4: Conclude the equality of corresponding angles. Since MNP PON, their corresponding angles are equal. The angle PMN in MNP corresponds to the angle NOP in PON. Therefore, PMN = NOP. This completes the proof. Proven that PMN = NOP