A line can be named after any two points it passes through. The line is indicated in green below.

The line does not pass through , so cannot be part of the name of the line. Specifically, is not a valid name.

A plane can be named after any three points on the plane that are not on the same line. , , and do not appear on the same line; for example, as can be seen below, the line that passes through and does not pass through .

Plane is a valid name for the plane that includes this figure.

A quadrilateral is named after its four vertices in consecutive order, going clockwise or counterclockwise. Quadrilateral is the figure in red, below:

, , , and are not a clockwise or counterclockwise ordering of the vertices, so Quadrilateral is not a valid name for the quadrilateral.

The sides of an angle are two rays, both of which begin at a common vertex. Since the middle letter of the name of this angle is , is the common endpoint; therefore, the name of each ray starts with . This makes and the correct choice.

Note that and cannot be correct: these are line segments.

A plane can be named after any three points on the plane that are not on the same line. As seen below, points , , and are on the same line.

Therefore, Plane is not a valid name for the plane.

The two sides of an angle are the two rays that compose it. Each of these rays begins at the vertex and proceeds out from there. In naming a ray, we always begin with the letter of the endpoint (where the ray starts) followed by another point on the ray in the direction it travels. Since the vertex of the angle is the endpoint of each ray and our vertex is , each of our rays must begin with . Only fails to do so.

A ray is defined by two points, one that defines the beginning, and the other that defines the direction - this ray goes on forever in whatever direction is defined by that second point

The two letters in the name of a line segment are its endpoints in either order. Therefore, is the segment with endpoints and ; consequently, it can also be called .

Two rays are opposite rays, by definition, if

(1) they have the same endpoint, and

(2) their union is a line.

The first letter in the name of a ray always refers to the endpoint of the ray. Therefore, has its endpoint at and has its endpoint at . The two rays are not opposite rays.

A ray is named with two letters, the first of which is an endpoint and the second of which is any other point the ray passes through. is a ray with endpoint , and is a ray with endpoint ; and are therefore not the same ray.