Question

in the right triangle above, what is the relation between angle ( x ) and ( y )?
a ( x = 90 - y )
b ( x = 90 + y )
c ( x = 180 - y )
d ( x = y )

Explanation:

Step1: Use the property of similar triangles

Since the two triangles (the large right - triangle and the small right - triangle) are similar. In a right - triangle, the sum of the two non - right angles is \(90^{\circ}\). Let's consider the large right - triangle \(\triangle ABC\) with right - angle at \(B\). The sum of angles in a triangle is \(180^{\circ}\). In \(\triangle ABC\), \(\angle A+\angle C = 90^{\circ}\). Also, note that the small right - triangle has an angle \(x\) and the relationship between the angles of the two similar triangles. Another way is to use the fact that the two non - right angles in a right - triangle are complementary.
Let's assume the two triangles (the big right - triangle and the small right - triangle) are similar. The angle \(x\) and the angle adjacent to \(y\) in the big triangle (excluding the right - angle) are equal. And since in the big right - triangle \(\angle A+\angle C=90^{\circ}\) and \(\angle C\) is related to \(y\) such that if we consider the angle sum property.
In a right - triangle, if one non - right angle is \(x\) and the other non - right angle (related to \(y\)):
We know that in a right - triangle \(\triangle ABC\) (\(\angle B = 90^{\circ}\)), \(\angle A+\angle C=90^{\circ}\). Also, from the similar triangles (the small right - triangle and \(\triangle ABC\)), we can see that \(x\) and the angle \((90 - y)\) are in a relationship.
Let's use the angle - angle similarity. The small right - triangle and the large right - triangle \(\triangle ABC\) have two right angles and share the angle \(x\). So, the other non - right angles are also related.
We know that in the large right - triangle \(\angle A+\angle C = 90^{\circ}\), and from the similar triangles, we can rewrite it in terms of \(x\) and \(y\).
Since the sum of angles in a right - triangle (the large one) is \(180^{\circ}\) (\(\angle A+\angle B+\angle C=180^{\circ}\), \(\angle B = 90^{\circ}\)), \(\angle A+\angle C=90^{\circ}\). Also, from the geometric relationship of the two triangles (using the fact that the two triangles are similar by AA (angle - angle) criterion: one right angle and the common angle \(x\)), we can say that \(x + y=90^{\circ}\) (because the non - right angles of a right - triangle are complementary).

Answer:

A. \(x = 90 - y\)