ALGEBRA \( W X Y Z \) is a kite. 49 If \( m \angle W X Y=120, m \angle W Z Y=4 x \), and \( m \angle Z W X=10 x \), find \( m \angle Z Y X \). 50. If \( m \angle W X Y=13 x+24, m \angle W Z Y=35 \), and \( m \angle Z W X=13 x+14 \), find \( m \angle Z Y X \).

Let's solve each problem step by step. --- ### **Problem 49** **Given:** - Quadrilateral \( WXYZ \) is a kite. - \( m \angle WXY = 120^\circ \) - \( m \angle WZY = 4x \) - \( m \angle ZWX = 10x \) **Find:** \( m \angle ZYX \) **Solution:** 1. **Understanding the Kite:** - In a kite, two pairs of adjacent sides are equal. - The sum of the interior angles of any quadrilateral is \( 360^\circ \). 2. **Identify Known Angles:** - \( \angle WXY \) is at vertex \( X \) and measures \( 120^\circ \). - \( \angle WZY \) is at vertex \( Z \) and measures \( 4x \). - \( \angle ZWX \) is at vertex \( W \) and measures \( 10x \). 3. **Calculate the Remaining Angle at Vertex \( Y \):** - Let \( \angle ZYX = y \). - Sum of all angles in the kite: \[ 10x + 120 + 4x + y = 360^\circ \] \[ 14x + y = 240^\circ \quad \Rightarrow \quad y = 240^\circ - 14x \] 4. **Apply Kite Property:** - In a kite, the angles at \( W \) and \( Y \) are equal. \[ y = 10x \] \[ 10x = 240^\circ - 14x \quad \Rightarrow \quad 24x = 240^\circ \quad \Rightarrow \quad x = 10 \] 5. **Find \( \angle ZYX \):** \[ y = 10x = 10 \times 10 = 100^\circ \] **Answer:** \( m \angle ZYX = 100^\circ \) --- ### **Problem 50** **Given:** - Quadrilateral \( WXYZ \) is a kite. - \( m \angle WXY = 13x + 24^\circ \) - \( m \angle WZY = 35^\circ \) - \( m \angle ZWX = 13x + 14^\circ \) **Find:** \( m \angle ZYX \) **Solution:** 1. **Understanding the Kite:** - In a kite, two pairs of adjacent sides are equal. - The sum of the interior angles of any quadrilateral is \( 360^\circ \). 2. **Identify Known Angles:** - \( \angle WXY \) is at vertex \( X \) and measures \( 13x + 24^\circ \). - \( \angle WZY \) is at vertex \( Z \) and measures \( 35^\circ \). - \( \angle ZWX \) is at vertex \( W \) and measures \( 13x + 14^\circ \). 3. **Calculate the Remaining Angle at Vertex \( Y \):** - Let \( \angle ZYX = y \). - Sum of all angles in the kite: \[ (13x + 14) + (13x + 24) + 35 + y = 360^\circ \] \[ 26x + 73 + y = 360^\circ \quad \Rightarrow \quad y = 287^\circ - 26x \] 4. **Apply Kite Property:** - In a kite, the angles at \( W \) and \( Y \) are equal. \[ y = 13x + 14 \] \[ 13x + 14 = 287 - 26x \quad \Rightarrow \quad 39x = 273 \quad \Rightarrow \quad x = 7 \] 5. **Find \( \angle ZYX \):** \[ y = 13x + 14 = 13 \times 7 + 14 = 105^\circ \] **Answer:** \( m \angle ZYX = 105^\circ \) --- ### **Final Answers:** - **Problem 49 Answer:** \( m \angle ZYX = 100^\circ \) - **Problem 50 Answer:** \( m \angle ZYX = 105^\circ \)