Question

1. find the value of ( x ) and the measure of ( angle a ).

Explanation:

Step1: Identify isosceles triangle

The triangle has two sides labeled \(5x\) and \(3x + 2\), and a vertex angle of \(130^\circ\). In an isosceles triangle, the two equal sides are opposite the equal base angles, so the two sides \(5x\) and \(3x + 2\) are equal. Thus, we set up the equation \(5x=3x + 2\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides of the equation \(5x=3x + 2\):
\(5x-3x=3x + 2-3x\)
\(2x=2\)
Divide both sides by 2:
\(x = \frac{2}{2}=1\)

Step3: Find the measure of \(\angle A\)

First, find the length of the equal sides. Substitute \(x = 1\) into \(3x+2\) (or \(5x\)): \(3(1)+2 = 5\) and \(5(1)=5\), so the two equal sides are 5 units. The sum of the interior angles of a triangle is \(180^\circ\). Let the measure of \(\angle A\) be \(y\), and since the two base angles (including \(\angle A\)) are equal (because the triangle is isosceles with two equal sides), we have \(2y+130^\circ=180^\circ\).
Subtract \(130^\circ\) from both sides: \(2y=180^\circ - 130^\circ=50^\circ\)
Divide both sides by 2: \(y=\frac{50^\circ}{2} = 25^\circ\)

Answer:

\(x = 1\), and the measure of \(\angle A\) is \(25^\circ\)