Question

in \\( \triangle p q r, overline{p q} cong overline{r p} \\) and \\( m angle p = 57 ^ { circ } \\). find \\( m angle r \\).

Explanation:

Step1: Identify the triangle type

Since \(\overline{PQ} \cong \overline{RP}\), \(\triangle PQR\) is isosceles with \(\angle R \cong \angle Q\) (base angles of an isosceles triangle are equal).

Step2: Use the triangle angle sum theorem

The sum of the interior angles of a triangle is \(180^\circ\). Let \(m\angle R = m\angle Q = x\). Then we have the equation:

$$ m\angle P + m\angle R + m\angle Q = 180^\circ $$

Substitute \(m\angle P = 57^\circ\) and \(m\angle R = m\angle Q = x\) into the equation:

$$ 57^\circ + x + x = 180^\circ $$

Step3: Solve for \(x\)

Simplify the equation:

$$ 57^\circ + 2x = 180^\circ $$

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:

\(m\angle R = 61.5^\circ\)