11. In \( \triangle W X Y \), if \( m \angle W \) is five less than twice \( m \angle Y \) and \( m \angle X \) is 21 more than \( m \angle Y \), find the measure of each angle.

Let’s determine the measures of the angles in triangle \( \triangle WXY \) using the given relationships. 1. **Define the Variables:** - Let \( m \angle Y = y \) degrees. - Then, according to the problem: - \( m \angle W = 2y - 5 \) degrees. - \( m \angle X = y + 21 \) degrees. 2. **Use the Triangle Angle Sum Property:** The sum of the angles in any triangle is \( 180^\circ \). Therefore: \[ m \angle W + m \angle X + m \angle Y = 180 \] Substitute the expressions for each angle: \[ (2y - 5) + (y + 21) + y = 180 \] 3. **Solve for \( y \):** Combine like terms: \[ 2y - 5 + y + 21 + y = 180 \] \[ 4y + 16 = 180 \] Subtract 16 from both sides: \[ 4y = 164 \] Divide by 4: \[ y = 41 \] 4. **Find Each Angle:** - \( m \angle Y = 41^\circ \) - \( m \angle W = 2(41) - 5 = 82 - 5 = 77^\circ \) - \( m \angle X = 41 + 21 = 62^\circ \) 5. **Verification:** \[ 77^\circ + 62^\circ + 41^\circ = 180^\circ \] The angles sum up correctly. **Final Answer:** - \( m \angle W = 77^\circ \) - \( m \angle X = 62^\circ \) - \( m \angle Y = 41^\circ \)