Question

1. the given figure shows two isosceles triangles within a rectangle. if ( b ) is half of ( c ), what is the value of ( a )?

Explanation:

Step1: Find the value of \(c\)

In an isosceles triangle with a \(36^{\circ}\) angle, using the angle - sum property of a triangle (\(180^{\circ}\)).
The two equal angles in the first isosceles triangle (with \(36^{\circ}\) angle) are \(\frac{180 - c}{2}\). But also, since it's part of the rectangle - related angles, we can use the formula \(c=180 - 2\times36\).

$$c = 180-72=108^{\circ}$$

Step2: Find the value of \(b\)

Given \(b=\frac{c}{2}\), substituting \(c = 108^{\circ}\)

$$b=\frac{108}{2}=54^{\circ}$$

Step3: Find the value of \(a\)

In the second isosceles triangle (where the two equal sides are marked), using the angle - sum property of a triangle (\(180^{\circ}\)) for the lower - left triangle in the rectangle.
Let's assume the sum of angles in the lower - left part of the rectangle (involving \(a\) and the isosceles triangle). The right - angle of the rectangle is \(90^{\circ}\).
We know that in the isosceles triangle with angle \(b = 54^{\circ}\), the other two equal angles (related to \(a\)): \(a=\frac{90 - b}{2}\)
Substituting \(b = 54^{\circ}\)

$$a=\frac{90 - 54}{2}=\frac{36}{2}=18^{\circ}$$

Answer:

\(18\)