In the given square PQRS, M and N are the points on the sides SR and QR respectively such that PM = SN O is the point of intersection of PM and SN. Prove that

(a) triangle PMS cong triangle RNS

(b) angle SOM = 90°

Given the square \( PQRS \), with points \( M \) and \( N \) on sides \( SR \) and \( QR \) respectively, such that \( PM = SN \), and point \( O \) is the intersection of \( PM \) and \( SN \), we need to prove the following:

### (a) Proving that \( \triangle PMS \cong \triangle RNS \)

1. **In \( \triangle PMS \) and \( \triangle RNS \):**

   - **Side \( PM = SN \)** (given).
   - **Side \( PS = RS \):** Since \( PQRS \) is a square, all sides are equal, so \( PS = RS \).
   - **Angle \( \angle PSM = \angle RSN \):** Both of these are right angles since they are angles formed at the corners of the square.

2. By the **SAS (Side-Angle-Side)** congruence rule, \( \triangle PMS \cong \triangle RNS \), because we have two sides equal and the angle between them also equal.

Thus, we have proven that \( \triangle PMS \cong \triangle RNS \).

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### (b) Proving that \( \angle SOM = 90^\circ \)

Since \( \triangle PMS \cong \triangle RNS \), the two triangles are mirror images of each other with respect to line \( OS \). This symmetry implies that lines \( PM \) and \( SN \) are perpendicular to each other at point \( O \).

To verify this:

1. From part (a), we know that \( \triangle PMS \cong \triangle RNS \).
2. The congruence implies that angles formed by the intersecting lines \( PM \) and \( SN \) are equal.
3. Since the lines are symmetrical with respect to the center \( O \), and the total angle around a point is \( 360^\circ \), each angle around \( O \) will be \( 90^\circ \).

Thus, \( \angle SOM = 90^\circ \), proving that \( PM \perp SN \).

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### Do you have any questions or would you like further details?

Here are 5 related questions to explore the topic further:

1. How do we find the coordinates of the points \( M \) and \( N \) given specific dimensions of the square?
2. How can we calculate the length of diagonals in a square?
3. What is the relationship between symmetry and congruent triangles?
4. How can we prove congruence using other methods like ASA or SSS?
5. What other properties of squares are useful in geometric proofs like this?

**Tip:** When dealing with square geometry, symmetry and congruence often simplify complex problems!

In the given square PQRS, M and N are the points on the sides SR and QR respectively such that PM = SN. O is the point of intersection of PM and SN. Prove that (a) triangle PMS is congruent to triangle RNS (b) angle SOM = 90°.

Explore a geometric proof involving a square PQRS where points M and N lie on sides SR and QR, respectively. Learn how to prove triangle PMS is congruent to triangle RNS using SAS congruence and that angle SOM equals 90°. This problem emphasizes concepts of symmetry, congruence, and right angles, suitable for students in Grades 8-10.

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