Question

the vertex angle of an isosceles triangle measures 42°. a base angle in the triangle has a measure given by (2x + 3)°. what is the value of x? what is the measure of each base angle? x = each base angle measures °.

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). In an isosceles triangle, the two base angles are equal. Let the vertex angle be \(42^\circ\) and each base angle be \((2x + 3)^\circ\). So the equation is \(42 + 2(2x + 3)=180\).

Step2: Simplify the equation

First, expand the left - hand side: \(42+4x + 6 = 180\). Then combine like terms: \(4x+48 = 180\).

Step3: Solve for x

Subtract 48 from both sides: \(4x=180 - 48=132\). Then divide both sides by 4: \(x=\frac{132}{4}=33\).

Step4: Find the measure of each base angle

Substitute \(x = 33\) into the expression for the base angle \((2x + 3)^\circ\). So \(2\times33+3=66 + 3=69^\circ\).

Answer:

\(x = 33\)
Each base angle measures \(69^\circ\)