Question

using algebra find the values of x and y.
23.

Explanation:

Step1: Analyze the equilateral triangle

Since all sides of the left - hand triangle are equal (marked with the same tick marks), it is an equilateral triangle. In an equilateral triangle, each angle is \(60^{\circ}\).

Step2: Find the value of \(x\)

The large triangle is divided into an equilateral triangle and another triangle. The angle adjacent to the right - angle (\(90^{\circ}\)) in the non - equilateral part: The angle of the equilateral triangle is \(60^{\circ}\). We know that the sum of angles in a triangle formed by the right - angle, \(x\) and part of the equilateral triangle angle.
The angle of the equilateral triangle is \(60^{\circ}\), and we consider the relationship between the angles. The non - equilateral triangle part: The angle adjacent to the right - angle (\(90^{\circ}\)) and the equilateral triangle's \(60^{\circ}\) angle.
We know that \(x=\frac{60^{\circ}}{2}=30^{\circ}\) (because of the property of the isosceles triangle formed in the non - equilateral part, as two sides are equal, so the base angles are equal and the vertex angle is \(60^{\circ}\) from the equilateral triangle's division).

Step3: Find the value of \(y\)

For the quadrilateral (the whole figure composed of the two triangles), the sum of interior angles of a quadrilateral is \(360^{\circ}\). One angle is \(90^{\circ}\) (right - angle), one is \(60^{\circ}\) (from the equilateral triangle), and we have two \(y\) angles (the two non - right, non - \(60^{\circ}\) angles in the non - equilateral triangle part, which are equal as the sides are equal).

$$360^{\circ}=90^{\circ}+60^{\circ}+2y$$

$$2y = 360^{\circ}-90^{\circ}-60^{\circ}=210^{\circ}$$

$$y = 105^{\circ}$$

Answer:

\(x = 30\), \(y=105\)