Question

ln and oq are parallel lines.
which angles are supplementary angles?
∠qpr and ∠lmk
∠qpm and ∠qpr
∠nmk and ∠qpm
∠lmk and ∠nmp

Explanation:

Step1: Recall Supplementary Angles Definition

Supplementary angles are two angles whose sum is \(180^\circ\) (a straight angle). They can be adjacent (forming a linear pair) or non - adjacent.

Step2: Analyze Each Option

• Option 1: \(\angle QPR\) and \(\angle LMK\)

Since \(LN\parallel OQ\) and \(KR\) is a transversal, \(\angle QPR\) and \(\angle LMK\) are corresponding angles. Corresponding angles are equal when lines are parallel, so their sum is not necessarily \(180^\circ\) (unless each is \(90^\circ\), which we don't know from the diagram). So they are not supplementary.

• Option 2: \(\angle QPM\) and \(\angle QPR\)

\(\angle QPM\) and \(\angle QPR\) form a linear pair (they are adjacent and their non - common sides form a straight line). By the definition of a linear pair, the sum of angles in a linear pair is \(180^\circ\). So \(\angle QPM+\angle QPR = 180^\circ\), which means they are supplementary. But let's check the other options to be sure.

• Option 3: \(\angle NMK\) and \(\angle QPM\)

\(\angle NMK\) and \(\angle QPM\): Since \(LN\parallel OQ\), \(\angle NMK\) and \(\angle QPM\) are alternate interior angles (if we consider the transversal). Alternate interior angles are equal, so their sum is not \(180^\circ\) (unless each is \(90^\circ\)). So they are not supplementary.

• Option 4: \(\angle LMK\) and \(\angle NMP\)

\(\angle LMK\) and \(\angle NMP\) are vertical angles. Vertical angles are equal, so their sum is not \(180^\circ\) (unless each is \(90^\circ\)). So they are not supplementary.

Answer:

\(\angle QPM\) and \(\angle QPR\) (the second option: \(\boldsymbol{\angle QPM}\) and \(\boldsymbol{\angle QPR}\))