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.

Sure, let's tackle each problem one by one. --- ### **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:** - \( m \angle S = 92^\circ \) - \( m \angle A = 92^\circ \) - \( m \angle L = 88^\circ \) - \( m \angle M = 88^\circ \) --- ### **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. --- I hope this helps! Let me know if you have any further questions.