Question

question
in \\( \triangle p q r, \overline{q r} \cong \overline{p q} \\) and \\( m \angle q = 84 ^ { \circ } \\). find \\( m \angle p \\).

Explanation:

Step1: Identify the triangle type

Since \(\overline{QR}\cong\overline{PQ}\), \(\triangle PQR\) is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. So \(\angle R=\angle P\).

Step2: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle P = x\) and \(m\angle R=x\) (because \(\angle P\cong\angle R\)), and \(m\angle Q = 84^{\circ}\). Then \(x + x+84^{\circ}=180^{\circ}\).

Step3: Solve the equation for \(x\)

Combine like terms: \(2x+84^{\circ}=180^{\circ}\). Subtract \(84^{\circ}\) from both sides: \(2x=180^{\circ}- 84^{\circ}=96^{\circ}\). Divide both sides by \(2\): \(x=\frac{96^{\circ}}{2}=48^{\circ}\).

Answer:

\(m\angle P = 48^{\circ}\)