Question

find the values of x and y.
x = 20
y = 50

Explanation:

Step1: Find the value of \(x\)

Since the two segments are equal, the base angles of the isosceles triangle are equal. Using the exterior - angle property of a triangle (an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles). But here, if we consider the small isosceles triangle, the base angles are equal. Let's use the property of right - angled triangles.
We know that in a right - angled triangle, if we consider the relationship between the angles. The angle adjacent to \(x\) (in the isosceles part) is \(x\) (base angles of isosceles triangle). And \(40^{\circ}=2x\) (exterior angle related to the two equal \(x\) angles in the small isosceles triangle part near the right - angle). So \(x = 20^{\circ}\) (already given correctly).

Step2: Find the value of \(y\)

In the large right - angled triangle, the sum of angles is \(180^{\circ}\). We know one angle is \(90^{\circ}\) and we can find the third angle.
The sum of angles in a triangle is \(180^{\circ}\). In the large right - angled triangle, one angle is \(90^{\circ}\), and another angle: we know from the small isosceles triangle part, the angle adjacent to the right - angle (excluding \(x\)) is \(2x=40^{\circ}\). So \(y+90^{\circ}+(40^{\circ}+x)=180^{\circ}\). Substitute \(x = 20^{\circ}\), we get \(y+90^{\circ}+60^{\circ}=180^{\circ}\).

$$y=180^{\circ}-(90^{\circ} + 60^{\circ})$$

$$y = 30^{\circ}$$

Answer:

\(x = 20\), \(y=30\)