12. If quadrilateral $$ S L A M $$ is a parallelogram and $$ m \angle S=92 $$, what are the measures of angles $$ L, A $$, and $$ M $$ ? 13. Prove that a diagonal and the sides of a parallelogram form two congru triangles.

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### **Problem 12**

**If quadrilateral $$ S L A M $$ is a parallelogram and $$ m \angle S = 92^\circ $$, what are the measures of angles $$ L $$, $$ A $$, and $$ M $$?**

**Solution:**

A **parallelogram** has specific properties regarding its angles:

1. **Opposite angles are equal.**
2. **Consecutive angles are supplementary** (i.e., their measures add up to $$ 180^\circ $$).

Given:
- $$ m \angle S = 92^\circ $$

Let's identify the positions of the angles in parallelogram $$ S L A M $$:

- **Angle S and Angle A** are opposite angles.
- **Angle L and Angle M** are opposite angles.

**Step-by-Step:**

1. **Find the measure of Angle A:**
   - Since opposite angles in a parallelogram are equal:
     $$
     m \angle A = m \angle S = 92^\circ
     $$

2. **Find the measures of Angles L and M:**
   - Consecutive angles are supplementary:
     $$
     m \angle S + m \angle L = 180^\circ \
     92^\circ + m \angle L = 180^\circ \
     m \angle L = 180^\circ - 92^\circ = 88^\circ
     $$
   - Since opposite angles are equal:
     $$
     m \angle M = m \angle L = 88^\circ
     $$

**Summary of Angle Measures:**
- $$ m \angle S = 92^\circ $$
- $$ m \angle A = 92^\circ $$
- $$ m \angle L = 88^\circ $$
- $$ m \angle M = 88^\circ $$

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### **Problem 13**

**Prove that a diagonal and the sides of a parallelogram form two congruent triangles.**

**Solution:**

To prove that the two triangles formed by a diagonal in a parallelogram are congruent, we'll use the **Side-Angle-Side (SAS)** congruence criterion.

**Given:**
- Quadrilateral $$ S L A M $$ is a parallelogram.
- Diagonal $$ S A $$ is drawn, forming two triangles: $$ \triangle S A M $$ and $$ \triangle S A L $$.

**To Prove:**
- $$ \triangle S A M \cong \triangle S A L $$

**Properties of a Parallelogram to Use:**
1. Opposite sides are equal in length.
2. Opposite angles are equal.
3. Consecutive angles are supplementary.
4. Diagonals bisect each other.

**Proof Using SAS Congruence:**

Let's consider diagonal $$ S A $$ as the common side.

1. **Side 1: Side $$ S A $$ is common to both triangles.**
   $$
   S A = S A \quad \text{(Reflexive Property)}
   $$

2. **Side 2: Side $$ S L $$ is equal to Side $$ S M $$.**
   - Since $$ S L A M $$ is a parallelogram, opposite sides are equal:
     $$
     S L = M A
     $$

3. **Angle Between the Sides:**
   - The angle between $$ S A $$ and $$ S L $$ in $$ \triangle S A L $$ is equal to the angle between $$ S A $$ and $$ S M $$ in $$ \triangle S A M $$. This is because opposite angles in a parallelogram are equal.
     $$
     m \angle S A L = m \angle S A M
     $$

**Applying SAS Criterion:**
- Since two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.

$$
\triangle S A L \cong \triangle S A M \quad \text{(SAS Postulate)}
$$

**Conclusion:**
- The diagonal $$ S A $$ in parallelogram $$ S L A M $$ divides it into two congruent triangles, $$ \triangle S A L $$ and $$ \triangle S A M $$, by the Side-Angle-Side congruence criterion.

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I hope this helps! Let me know if you have any further questions.
In parallelogram $$ S L A M $$: - $$ m \angle S = 92^\circ $$ - $$ m \angle A = 92^\circ $$ - $$ m \angle L = 88^\circ $$ - $$ m \angle M = 88^\circ $$ **Proof:** A diagonal in a parallelogram divides it into two congruent triangles by the Side-Angle-Side (SAS) congruence criterion.

12. If quadrilateral \( S L A M \) is a parallelogram and \( m \angle S=92 \), what are the measures of angles \( L, A \), and \( M \) ? 13. Prove that a diagonal and the sides of a parallelogram form two congru triangles.

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