Question

△abc is an isosceles triangle, where \overline{ab} \cong \overline{ac}.

the measure of \angle c is 45^{circ}. what are the measures of the missing angles?
m\angle a

Explanation:

Step1: Use the property of isosceles triangle

In an isosceles triangle \( \triangle ABC\) with \( \overline{AB}\cong\overline{AC}\), the base - angles are equal. So \( \angle B=\angle C\). Given \( m\angle C = 45^{\circ}\), then \( m\angle B=45^{\circ}\).

Step2: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \( 180^{\circ}\). Let \( m\angle A=x\). Then \( x + m\angle B+m\angle C=180^{\circ}\). Substitute \( m\angle B = 45^{\circ}\) and \( m\angle C = 45^{\circ}\) into the equation: \( x+45^{\circ}+45^{\circ}=180^{\circ}\).

Step3: Solve for \( x\)

Simplify the left - hand side of the equation: \( x + 90^{\circ}=180^{\circ}\). Subtract \( 90^{\circ}\) from both sides: \( x=180^{\circ}-90^{\circ}\).

Answer:

\( m\angle A = 90^{\circ}\)