Question

1. \\( \angle b = \\)______

find the missing angle measures.
\\( \angle c = \\)______

1. \\( x = \\)______

find the value of x.

Explanation:

Step1: Identify the triangle type for problem 1

The triangle has two equal - length sides (marked with the same tick marks), so it is an isosceles triangle. In an isosceles triangle, the base angles (\(\angle B\) and \(\angle C\)) are equal.

Step2: Use the angle - sum property of a triangle for problem 1

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(\angle B=\angle C = x\). Then \(x + x+72^{\circ}=180^{\circ}\), which simplifies to \(2x=180^{\circ}-72^{\circ}=108^{\circ}\). So \(x = 54^{\circ}\). Thus \(\angle B = 54^{\circ}\) and \(\angle C=54^{\circ}\).

Step3: Identify the triangle type for problem 2

The triangle has two equal angles (\(\angle B\) and \(\angle C\)), so it is an isosceles triangle. In an isosceles triangle, the sides opposite the equal angles are equal. So \(4x + 5=37\).

Step4: Solve the equation for problem 2

Subtract 5 from both sides of the equation \(4x+5 = 37\): \(4x=37 - 5=32\). Then divide both sides by 4: \(x=\frac{32}{4}=8\).

Answer:

1. \(\angle B = 54^{\circ}\), \(\angle C = 54^{\circ}\)
2. \(x = 8\)