can be seen to be completely contained in  - that is, . The intersection of a set and its subset is the subset, so the correct response is .

The diagram below show  and  in red and green, respectively:

The intersection of  and  is the set of points they have in common, which can be seen to be the portion of the line with endpoints  and . This figure is the line segment .

A ray is named after its endpoint and any other point on the ray, in that order. Since  is the first letter in the name ,  is its endpoint, and any other name for the ray must begin with ; this allows us to eliminate . Also,  is eliminated, since a ray is named after two, not three, points.  is a correct choice, since the first letter in this name is endpoint , and the second letter names a point on this ray.

Below is the diagram with the points , and , as described, shown in green. Also, , the ray that has endpoint  and passes through , is marked in red.

The ray also passes through , , and , so ,  , and  are also valid names for the ray. However, the ray does not pass through , so  is not a valid name for the ray.  is the correct choice.

Opposite rays are rays that have the same endpoint are on the same line, but share only the endpoint. 

Two names of opposite rays must begin with the same letter, since the first letter in the name of a ray is the endpoint. Also, the second letters must represent points collinear with each other and the endpoint, and on opposite sides of the shared endpoint. Of the five choices, only  and  together meet this criteria.

 and  are noncollinear rays with the same endpoint, so their union is an angle. The shared endpoint of the rays, which here is , is the vertex of the formed angle, and is therefore the middle letter of the name of the angle; this letter is flanked by two letters representing one point on each ray, which are  and . The correct choice among those given is .

 can be seen to be completely contained in  - that is, . The union of a set and its subset is the containing set, so the correct response is .

The diagram below shows  and  in red and green, respectively:

The union of the two sets is the set of points in one or the other; this set is the entire line containing the two rays, which is .

Below is the diagram with midpoint  of  added; also, , the ray starting at  and passing through , is in green.

A ray is named after, in order, its endpoint and any other point on the ray.   has  as an endpoint, and also includes the points  and , so there are two valid alternative names,  and , among the choices. The correct response is .

Below is the diagram with the points , and , as described, shown in green. Also, , the ray that has endpoint  and passes through , is marked in red.

The ray also passes through  and , so , , , and —all four given names—are also valid names for the ray.