Question

exercise #4: the ratio of the measure of either base angle to the measure of the vertex angle in an isosceles triangle is 3:2. determine the measure of the vertex angle of this isosceles triangle.

Explanation:

Step1: Set up the equation based on angle sum property

Let the measure of the vertex angle be \(3x\) and the measure of each base angle be \(2x\). In a triangle, the sum of interior angles is \(180^{\circ}\). So, \(3x + 2x+2x=180^{\circ}\).

Step2: Solve for \(x\)

Combine like - terms: \(7x = 180^{\circ}\), then \(x=\frac{180^{\circ}}{7}\approx25.71^{\circ}\). But wait, we made a wrong assumption. Since \(A = 66^{\circ}\) is a base angle. Let the vertex angle be \(y\). The ratio of base angle to vertex angle is \(2:3\). So \(\frac{66}{y}=\frac{2}{3}\).
Cross - multiply: \(2y=66\times3\).

Step3: Solve for the vertex angle \(y\)

\(2y = 198\), then \(y=\frac{198}{2}=99^{\circ}\).

Answer:

The measure of the vertex angle is \(99^{\circ}\)