For two rays to be the same, they must have the same endpoint and point in the same direction. Let's look at each of the pairs of rays one at a time and see if they meet these conditions.

$K P$ and $P K$

The ray $K P$ has the endpoint $K,$ goes through the point $P,$ and continues indefinitely. Note that point $P$ is the intersection between the two lines $K L$ and $M N .$ Let's highlight $K P$ and $P$ in the diagram to make it easier for us to visualize the ray.

What about the ray $P K ?$ It has the endpoint $P,$ goes through the point $K,$ and continues on indefinitely. Since they have different endpoints and point in different directions, we can draw the conclusion that this ray is not the same as $K P .$ We can highlighted $P K$ in the diagram to make the differences more apparent.

$N P$ and $N M$

$N P$ has the endpoint $N,$ goes through $P,$ and continues indefinitely. Let's highlight this ray in the diagram.

$N M$ also has the endpoint $N .$ Remember, the two rays are only the same if they also point in the same direction. $N M$ must travel through point $P$ on its way to go through the point $M .$ This can be seen on the diagram above. We can know for sure that this is the case since the points $N,$ $P,$ and $M$ are all collinear. This means that the two rays are indeed pointing in the same direction, and are, therefore, the same.

KP and PK

NP and NM

$K P$ and $P K$

$N P$ and $N M$

Exercise