Question

1. the measure of the vertex angle of an isosceles triangle is 12 more than 5 times themeasure of a base angle. determine the sum of the measures of the base angles.

Explanation:

Step1: Let the measure of a base angle be \(x\)

Let the measure of a base angle be \(x\). Then the measure of the vertex angle is \(5x + 12\).

Step2: Use the angle - sum property of a triangle

In an isosceles triangle, the sum of the interior angles is \(180^{\circ}\). So, \(x+x+(5x + 12)=180\).
Simplify the left - hand side: \(7x+12 = 180\).
Subtract 12 from both sides: \(7x=180 - 12=168\).
Divide both sides by 7: \(x = 24\).

Step3: Find the sum of the base angles

The sum of the base angles is \(2x\). Substitute \(x = 24\) into \(2x\), we get \(2\times24=48\).

Answer:

The sum of the measures of the base angles is \(48^{\circ}\).