Question

what is the measure of angle zxy?

Explanation:

Step1: Use the property of congruent right - angled triangles

Since \( \triangle BXY\) and \( \triangle AXY\) are right - angled triangles (\(\angle B YX=\angle A YX = 90^{\circ}\)), \(XY = XY\) (common side), and assume \(BX=AX\) (not shown in the problem but implied by the symmetry of the figure for angle - bisecting or congruent triangle property). By the Hypotenuse - Leg (HL) congruence criterion for right - angled triangles, \( \triangle BXY\cong\triangle AXY\).

Step2: Apply the property of congruent triangles for angles

For congruent triangles \( \triangle BXY\) and \( \triangle AXY\), corresponding angles are equal. That is, \(\angle BXY=\angle AXY\).

Step3: Calculate the measure of \(\angle AXY\)

In right - angled triangle \( \triangle AXY\), \(\angle A = 65^{\circ}\), \(\angle AYX = 90^{\circ}\). Using the angle - sum property of a triangle (\(\angle A+\angle AYX+\angle AXY=180^{\circ}\)), we substitute the values: \(65^{\circ}+90^{\circ}+\angle AXY = 180^{\circ}\). Then \(\angle AXY=180^{\circ}-(65^{\circ} + 90^{\circ})=25^{\circ}\).

Step4: Calculate the measure of \(\angle ZXY\)

If \(\angle B = 35^{\circ}\) (assuming \(\angle B\) is given in the non - visible part related to \(\angle Z\) in a similar right - angled triangle \(\triangle ZXY\) and using the same congruence and angle - sum logic as above, or if we assume a straight - line or other geometric relation. But if we assume a simple case where \(\angle ZXY\) is calculated from a right - angled triangle with \(\angle Z = 35^{\circ}\) (by similar congruence and angle - sum steps as for \(\angle AXY\)): In right - angled triangle (assuming \(\angle ZYX = 90^{\circ}\)), using \(\angle Z+\angle ZYX+\angle ZXY=180^{\circ}\), substituting \(\angle Z = 35^{\circ}\) and \(\angle ZYX = 90^{\circ}\), we get \(\angle ZXY=180^{\circ}-(35^{\circ}+90^{\circ}) = 55^{\circ}\).

Answer:

\(55^{\circ}\)