Question

question
in △opq, \overline{op} \cong \overline{qo} and m∠q = 48°. find m∠o.

Explanation:

Step1: Identify the triangle type

Since \(\overline{OP}\cong\overline{QO}\), \(\triangle OPQ\) is an isosceles triangle. In an isosceles triangle, the base - angles are equal. So, \(\angle P=\angle Q\). Given \(m\angle Q = 48^{\circ}\), then \(m\angle P=48^{\circ}\).

Step2: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle O=x\). Then, by the triangle - angle sum theorem \(x + m\angle P+m\angle Q=180^{\circ}\).
Substitute \(m\angle P = 48^{\circ}\) and \(m\angle Q = 48^{\circ}\) into the equation: \(x+48^{\circ}+48^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(x + 96^{\circ}=180^{\circ}\).
Solve for \(x\): \(x=180^{\circ}-96^{\circ}\).

Answer:

\(m\angle O = 84^{\circ}\)