Question

question
in △nop, \overline{op} \cong \overline{no} and m∠o = 57°. find m∠n.

Explanation:

Step1: Identify the triangle type

Since \(\overline{OP}\cong\overline{NO}\), \(\triangle NOP\) is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. So \(\angle P=\angle N\).

Step2: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle N = x\) and \(m\angle P=x\) (because \(\angle N\cong\angle P\)), and \(m\angle O = 57^{\circ}\). Then \(x + x+57^{\circ}=180^{\circ}\).
Combine like terms: \(2x+57^{\circ}=180^{\circ}\).
Subtract \(57^{\circ}\) from both sides: \(2x=180^{\circ}- 57^{\circ}=123^{\circ}\).
Divide both sides by 2: \(x=\frac{123^{\circ}}{2}=61.5^{\circ}\).

Answer:

\(m\angle N = 61.5^{\circ}\)