The two pairs of adjacent angles in the diagram are ∠K and ∠M, and ∠N and ∠O. The two pairs of vertical angles in the diagram are ∠K and ∠O, and ∠M and ∠N.
In the given diagram, there are two pairs of adjacent angles and two pairs of vertical angles.
Adjacent angles:∠K and ∠M ,∠N and ∠O
Adjacent angles share a common vertex and a common side.
In this diagram, ∠K and ∠M share the common vertex K and the common side KM, while ∠N and ∠O share the common vertex O and the common side ON.
Vertical angles:∠K and ∠O ,∠M and ∠N
Vertical angles are opposite angles formed when two lines intersect.
In this diagram, ∠K and ∠O are vertical angles because they are opposite each other when lines KM and ON intersect at point O.
Similarly, ∠M and ∠N are vertical angles because they are opposite each other when lines KM and ON intersect at point O.
Detailed explanation: The diagram shows two parallel lines, KM and ON, intersected by a transversal line, LO.
This creates four angles: ∠K, ∠M, ∠N, and ∠O.
Adjacent angles: Adjacent angles share a common vertex and a common side.
In this diagram, ∠K and ∠M share the common vertex K and the common side KM, while ∠N and ∠O share the common vertex O and the common side ON.
Vertical angles: Vertical angles are opposite angles formed when two lines intersect.
In this diagram, ∠K and ∠O are vertical angles because they are opposite each other when lines KM and ON intersect at point O.
Similarly, ∠M and ∠N are vertical angles because they are opposite each other when lines KM and ON intersect at point O.
Another way to identify adjacent and vertical angles is to use the concept of congruent angles.
Congruent angles are angles that have the same measure.
Adjacent angles are never congruent, but vertical angles are always congruent.
In other words, if two angles are vertical angles, then their measures will always be the same.
In the given diagram, we can see that ∠K and ∠O are vertical angles because they are opposite each other when lines KM and ON intersect.
Since vertical angles are always congruent, we know that ∠K and ∠O must have the same measure.
Similarly, we can see that ∠M and ∠N are vertical angles because they are opposite each other when lines KM and ON intersect.
Since vertical angles are always congruent, we know that ∠M and ∠N must have the same measure.
Therefore, the two pairs of adjacent angles in the diagram are ∠K and ∠M, and ∠N and ∠O.
The two pairs of vertical angles in the diagram are ∠K and ∠O, and ∠M and ∠N.