To find the measures of \(\angle X\) and \(\angle Y\) in the quadrilateral XYZW, we need to use the information provided in the picture details. Let's go through the steps to solve for these angles.Step 1: Analyze the given informationWe know that:- \(\angle XZY = 164°\)- \(\angle ZVW = 102°\)- \(\angle XWV\) is a right angle, so \(\angle XWV = 90°\)- \(\angle ZVY\) is a right angle, so \(\angle ZVY = 90°\)Step 2: Use the properties of quadrilateralsThe sum of the interior angles of any quadrilateral is 360°. Therefore, we can write the following equation:\(\angle X + \angle Y + \angle Z + \angle W = 360°\)Step 3: Find \(\angle Z\) and \(\angle W\)Since \(\angle XZY\) and \(\angle ZVY\) are adjacent angles that form a straight line, their sum is 180°. Therefore:\(\angle Z = 180° - \angle XZY = 180° - 164° = 16°\)Similarly, since \(\angle ZVW\) and \(\angle XWV\) are adjacent angles that form a straight line, their sum is also 180°. Therefore:\(\angle W = 180° - \angle ZVW = 180° - 102° = 78°\)Step 4: Substitute \(\angle Z\) and \(\angle W\) into the quadrilateral angle sum equationNow we can substitute the values of \(\angle Z\) and \(\angle W\) into the quadrilateral angle sum equation:\(\angle X + \angle Y + 16° + 78° = 360°\)Step 5: Solve for \(\angle X + \angle Y\)\(\angle X + \angle Y = 360° - 16° - 78°\)\(\angle X + \angle Y = 360° - 94°\)\(\angle X + \angle Y = 266°\)Step 6: Find \(\angle X\) and \(\angle Y\)We cannot determine the exact measures of \(\angle X\) and \(\angle Y\) with the given information alone because there are multiple possibilities for their measures that would add up to 266°. Additional information, such as the lengths of the sides or other angle measures, would be necessary to find the exact measures of \(\angle X\) and \(\angle Y\).Final Answer:Without additional information, we can only determine that the sum of \(\angle X\) and \(\angle Y\) is 266°. The exact measures of \(\angle X\) and \(\angle Y\) cannot be determined from the given information.
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