1. **Problem statement:** Given rectangle WXYZ with $m\angle XZY = 63^\circ$, find $m\angle WYX$ and $m\angle WVZ$. The diagonals WY and XZ intersect at point V. 2. **Recall properties of rectangles:** - All angles in a rectangle are right angles ($90^\circ$). - Diagonals of a rectangle are equal in length and bisect each other. - The diagonals form two congruent triangles. 3. **Analyze $m\angle XZY = 63^\circ$:** - Point Z is the top-right corner, Y is bottom-right, and X is bottom-left. - Triangle XZY is formed by points X, Z, and Y. - Since WXYZ is a rectangle, $\angle Z = 90^\circ$. 4. **Find $m\angle WYX$:** - $\angle WYX$ is at point Y between points W and X. - Triangle WYX is right-angled at Y (since rectangle corners are $90^\circ$). - The diagonal WY splits the rectangle into two congruent triangles. 5. **Use triangle XZY:** - Triangle XZY has angles $m\angle XZY = 63^\circ$, $m\angle ZXY$, and $m\angle XYZ$. - Since $\angle Z = 90^\circ$, the other two angles sum to $90^\circ$. - So, $m\angle ZXY + m\angle XYZ = 27^\circ$. 6. **By symmetry, $m\angle WYX = m\angle XYZ$:** - Because triangles WYX and XZY are congruent by diagonal symmetry. - Therefore, $m\angle WYX = 27^\circ$. 7. **Find $m\angle WVZ$:** - Point V is the intersection of diagonals WY and XZ. - Diagonals bisect each other, so V is midpoint of both. - $\angle WVZ$ is the angle between points W, V, and Z. 8. **Diagonals in rectangle:** - Diagonals are equal and bisect each other. - The diagonals form two congruent triangles at V. 9. **Calculate $m\angle WVZ$:** - Since $m\angle XZY = 63^\circ$, the diagonals intersect at $90^\circ - 63^\circ = 27^\circ$. - The angle between diagonals at V is twice this angle, so $m\angle WVZ = 2 \times 27^\circ = 54^\circ$. **Final answers:** $$m\angle WYX = 27^\circ$$ $$m\angle WVZ = 54^\circ$$