Learn to prove it,not just to compute it.

Answer

Full solution

1. Recall what a ray is: one endpoint, then a straight path that never ends in one direction. The first letter in the name tells you where the ray starts. The second letter is just any other point the ray passes through.

2. Test it on a number line. Put $P$ at $0$ and $Q$ at $3$. Then $\overrightarrow{PQ}$ is every point with coordinate $0$ or greater, and $\overrightarrow{QP}$ is every point with coordinate $3$ or less.

3. These two sets are different. The point at $5$ is on $\overrightarrow{PQ}$ but not on $\overrightarrow{QP}$, and the point at $-2$ is on $\overrightarrow{QP}$ but not on $\overrightarrow{PQ}$. One counterexample is enough to show the names do not describe the same ray.

4. Compare with the other names. $\overline{PQ}$ and $\overline{QP}$ are the same segment, and $\overleftrightarrow{PQ}$ and $\overleftrightarrow{QP}$ are the same line, because neither has a preferred starting point. Only a ray cares about the order of the letters.

5. One more detail worth knowing: these two rays are not opposite rays either, because they do not share an endpoint. They overlap along the whole segment $\overline{PQ}$.

Answer

Full solution

1. Test each candidate against two pictures. Picture 1 is the standard pair: $\overrightarrow{OA}$ and $\overrightarrow{OB}$ with $O$ between $A$ and $B$ on a line. Those are opposite rays, so a good definition must accept this pair. Picture 2 is any other pair that is not opposite rays, and a good definition must reject it.

2. Choice (A) is too loose. Two parallel rays in a plane can point in opposite directions without being on the same line, and they are not opposite rays. Choice (B) is too loose in another way: two rays that form an angle share an endpoint but do not make a line.

3. Choice (C) fails on a number line. The ray from $0$ toward the right and the ray from $3$ toward the left together cover the entire line, but they overlap on the segment from $0$ to $3$ and do not share an endpoint, so they are not opposite rays.

4. Choice (E) fails for the same reason as (B): two rays forming an angle meet only at their endpoint, and their union is a bent figure, not a line.

5. Choice (D) requires both a common endpoint (rules out the overlap in (C) and the parallel pair in (A)) and a union that is a line (rules out the angle in (B) and (E)). It is the only choice that accepts opposite rays and rejects every impostor.