Question

the measure of \\( \angle a \\) is \\( 118 ^ { \circ } \\). what are the measures of the missing angles?

\\( m \angle b \\)

\\( m \angle b = 31 \\)

\\( m \angle c \\)

\\( m \angle c = \\)

Explanation:

Step1: Use the property of isosceles triangles

In an isosceles triangle, if two sides are equal (marked with the same tick marks), then the angles opposite those sides are equal. Here, since the sides opposite \(\angle B\) and \(\angle C\) are equal, \(m\angle B=m\angle C\).

Step2: Use the triangle - angle sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle B = m\angle C=x\). We know that \(m\angle A=118^{\circ}\). Then, by the angle - sum formula \(m\angle A + m\angle B+m\angle C = 180^{\circ}\), we substitute the values: \(118^{\circ}+x + x=180^{\circ}\).

Step3: Solve the equation for \(x\)

Simplify the equation \(118 + 2x=180\). Subtract \(118\) from both sides: \(2x=180 - 118\). So, \(2x = 62\). Divide both sides by \(2\): \(x=\frac{62}{2}=31\).

Answer:

\(m\angle C = 31^{\circ}\)