Before we take a look at the given diagram, let's start by defining adjacent, complementary, supplementary, and vertical angles.

Type of angle	Description
Adjacent	Two angles that share a common vertex and side.
Complementary	Two angles whose measures have a sum of $9 0^{\circ} .$
Supplementary	Two angles whose measures have a sum of $18 0^{\circ} .$
Vertical	Opposite angles formed by an intersection of two lines.

Now, let's consider the given diagram.

To determine a pair of adjacent angles, let's take a look at the given diagram, looking for a pair of angles that share a $vertex$ and a $common$ $side .$

Looking at the diagram, $∠ L J K$ and $∠ K J Q$ share $J$ as a vertex and $J K$ as a side. Therefore, these angles are adjacent. Now, let's look for a pair of angles whose measures sum to $9 0^{\circ}$ — a pair of complementary angles.

Looking at the markings on the given diagram, $∠ M J P$ is a right angle — its measure is $9 0^{\circ} .$ As $∠ M J N$ and $∠ N J P$ are adjacent angles that sum to $∠ M J P,$ they are complementary. Now, let's look for a pair of angles that sum to $18 0^{\circ}$ — a pair of supplementary angles.

Looking at the markings on the given diagram, $∠ L J K$ and $∠ K J P$ sum to the straight angle $∠ L J P .$ Therefore, the sum of the measures of these angles is the measure of $∠ L J P$ — $18 0^{\circ} .$ As a result, $∠ L J K$ and $∠ K J P$ are supplementary. Finally, let's look for a pair of vertical angles. These angles are on the opposite sides of an intersection of two lines.

Looking at the diagram, $∠ L J K$ and $∠ P J N$ are on the opposite sides of an intersection of $L P$ and $N K .$ Therefore, these angles are vertical.

Exercise

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