Question

triangle opq is formed by connecting the midpoints of the side of triangle lmn. the measures of the interior angles of triangle lmn are shown. find the measure of ∠loq. figures not necessarily drawn to scale.

Explanation:

Step1: Find the measure of \(\angle LNM\)

In \(\triangle LMN\), using the angle - sum property of a triangle (\(\angle L+\angle M+\angle N = 180^{\circ}\)). Given \(\angle L = 49^{\circ}\), \(\angle M=49^{\circ}\), \(\angle N = 82^{\circ}\).

Step2: Use the mid - point theorem (mid - segment theorem)

Since \(O\), \(P\), \(Q\) are mid - points of the sides of \(\triangle LMN\), \(QO\parallel MN\) and \(LOQN\) is a parallelogram (a quadrilateral with both pairs of opposite sides parallel).
In a parallelogram, opposite angles are equal.
\(\angle LOQ+\angle N=180^{\circ}\) (adjacent angles of a parallelogram are supplementary)

Answer:

\(98^{\circ}\)