In Exercises 17 -20, find the measures of angle X and angle Y 17.

To find the measures of angles X and Y in the quadrilateral XYZW, we need to use the information given in the picture details and the properties of quadrilaterals.Step 1: Sum of angles in a quadrilateralThe sum of the interior angles in any quadrilateral is 360°. We can use this information to set up an equation to solve for the unknown angles.Step 2: Set up the equationLet's denote the measure of angle X as m∠X and the measure of angle Y as m∠Y. According to the picture details and the sum of angles in a quadrilateral, we have:m∠X + m∠Y + 164° + 90° = 360°Step 3: Simplify the equationNow, we combine the known angles:m∠X + m∠Y + 254° = 360°Step 4: Solve for m∠X + m∠YSubtract 254° from both sides of the equation:m∠X + m∠Y = 360° - 254°m∠X + m∠Y = 106°Step 5: Use the given angle to find m∠YWe know that ∠ZVW is 102°, and since ∠ZVW and ∠Y are supplementary (because they form a straight line), their measures add up to 180°. So we can find m∠Y by subtracting the measure of ∠ZVW from 180°:m∠Y = 180° - 102°m∠Y = 78°Step 6: Use the value of m∠Y to find m∠XNow that we know m∠Y, we can substitute it back into the equation from Step 4 to find m∠X:m∠X + 78° = 106°Subtract 78° from both sides:m∠X = 106° - 78°m∠X = 28°Final Answer:The measure of ∠X is 28°, and the measure of ∠Y is 78°.