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.

The first letter in the name of a ray refers to its endpoint; the second refers to the name of any other point on the ray.  and  are rays that have endpoint  and pass through  and , respectively. Those rays are indicated below in red and green, respectively.

As it turns out, the two rays are one and the same.

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 refers to its endpoint; the second refers to the name of any other point on the ray.  and  both have endpoint , so the first criterion is met.  passes through point  and  passes through point ;  and  are indicated below in green and red, respectively:

The union of the two rays is a line. Both criteria are met, so the rays are indeed opposite.

Three points are collinear if there is a single line that passes through all three. In the diagram below, it can be seen that the line that passes through  and  does not pass through .

Therefore, the three points are not collinear.

Three points are collinear if there is a single line that passes through all three. In the diagram below, it can be seen that such a line exists.

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.

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.

If  and  , then , since two lines parallel to the same line are parallel to each other.

If , then , since two same-side interior angles formed by transversal  are supplementary.

If , then , since two alternate interior angles formed by transversal  are congruent.

However,  regardless of whether  and  are parallel; they are vertical angles, and by the Vertical Angles Theorem, they must be congruent.

By the Isosceles Triangle Theorem, two interior angles must be congruent. However, since a triangle cannot have two obtuse interior angles, the two missing angles must be the ones that are congruent. Since the total angle measure of a triangle is  , each of the missing angles measures .