Question

in △mno, \\( \overline {mn}\cong \overline {om} \\) and \\( m\angle o = 33 ^ { \circ } \\). find \\( m\angle m \\).

Explanation:

Step1: Identify the triangle type

Since \( \overline{MN}\cong\overline{OM}\), \(\triangle MNO\) is an isosceles triangle. In an isosceles triangle, the base - angles are equal. So, \(m\angle N=m\angle O = 33^{\circ}\).

Step2: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle M=x\). Then, by the triangle - angle sum theorem \(x + m\angle N+m\angle O=180^{\circ}\).
Substitute \(m\angle N = 33^{\circ}\) and \(m\angle O = 33^{\circ}\) into the equation: \(x+33^{\circ}+33^{\circ}=180^{\circ}\).

Step3: Solve for \(x\)

Simplify the left - hand side of the equation: \(x + 66^{\circ}=180^{\circ}\).
Subtract \(66^{\circ}\) from both sides: \(x=180^{\circ}-66^{\circ}\).
\(x = 114^{\circ}\).

Answer:

\(m\angle M = 114^{\circ}\)