Question

if \\( \triangle m a p \cong \triangle t o n \\), \\( m \angle m = 80 ^ { \circ } \\), and \\( m \angle n = 50 ^ { \circ } \\), find \\( m \angle a \\).

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle MAP\cong\triangle TON\), then \(\angle A\) corresponds to \(\angle O\), \(\angle M\) corresponds to \(\angle T\), and \(\angle P\) corresponds to \(\angle N\). So \(m\angle P = m\angle N=50^{\circ}\).

Step2: Apply the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle MAP\), we know that \(m\angle M + m\angle A+m\angle P=180^{\circ}\).
Substitute \(m\angle M = 80^{\circ}\) and \(m\angle P = 50^{\circ}\) into the equation: \(80^{\circ}+m\angle A + 50^{\circ}=180^{\circ}\).

Step3: Solve for \(m\angle A\)

Combine like terms: \(m\angle A+130^{\circ}=180^{\circ}\).
Subtract \(130^{\circ}\) from both sides: \(m\angle A=180^{\circ}- 130^{\circ}\).

Answer:

\(m\angle A = 50^{\circ}\)