Question

in $\triangle opq, \overline{op} \cong \overline{qo}$ and $m\angle q = 48^{\circ}$. find $m\angle o$.

Explanation:

Step1: Identify triangle type

Since $\overline{OP} \cong \overline{QO}$, $\triangle OPQ$ is isosceles with base $\overline{PQ}$. The base angles opposite the congruent sides are $\angle P$ and $\angle Q$.

Step2: Find $m\angle P$

Base angles of isosceles triangle are equal, so $m\angle P = m\angle Q = 48^\circ$.

Step3: Calculate $m\angle O$

Use triangle angle sum theorem: $m\angle O + m\angle P + m\angle Q = 180^\circ$.
Substitute values:
$m\angle O = 180^\circ - 48^\circ - 48^\circ$
$m\angle O = 84^\circ$

Answer:

$84^\circ$