Question

in \\( \triangle opq, \overline{op} \cong \overline{qo} \\) and \\( m\angle q = 48 ^ { \circ } \\). find \\( m\angle 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, \(\angle P=\angle Q = 48^{\circ}\).

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 angle - sum formula \(x + m\angle P+m\angle Q=180^{\circ}\).
Substitute \(m\angle P = 48^{\circ}\) and \(m\angle Q = 48^{\circ}\) into the equation: \(x+48^{\circ}+48^{\circ}=180^{\circ}\).

Step3: Solve for \(x\)

Simplify the left - hand side of the equation: \(x + 96^{\circ}=180^{\circ}\).
Subtract \(96^{\circ}\) from both sides: \(x=180^{\circ}-96^{\circ}\).

Answer:

\(84^{\circ}\)