The name of a ray includes two letters, so  can be eliminated.

The first letter must be the endpoint. Since  is a name of the ray, the endpoint is , and any alternative name for the ray must begin with . This leaves only .

 and  are two acute angles of a right triangle and are therefore complementary - that is, 

, so

 and , being alternate interior angles formed by transversal  across parallel lines, are congruent, so .

We now look at , whose interior angles must have degree measures totaling , so

From the Parallel Postulate and its converse, as well as its various resulting theorems, two lines in a plane crossed by a transversal are parallel if any of the following happen:

Both lines are perpendicular to the same third line - this happens if  is a right angle, since, from this fact and the fact that  is also right, both lines are perpendicular to .

Same-side interior angles are supplementary - this happens if  and  are supplementary, since they are same-side interior angles with respect to transversal .

Alternate interior angles are congruent - this happens if , since they are alternate interior angles with respect to transversal .

However, the fact that   bisects  has no bearing on whether  is true or not, since it does not relate any two angles formed by a transversal. 

"  bisects " is the correct choice.

In an intersecting line, vertical angles are equal to each other.

Set up an equation such that both angles are equal.

Solve for .  Subtract  on both sides.

Add 14 on both sides.

Divide by 7 on both sides.

The answer is:  

Vertical angles are equal.

Set both angles equal and solve for x.

Subtract  on both sides.

Add 8 on both sides.

Divide by 4 on both sides.

The answer is:  

Vertical angles of intersecting lines must equal to each other.

Set up an equation such that both angle measures are equal.

Add three on both sides.

Divide by three on both sides.

The answer is:  

Opposite angles equal.  Set up an equation such that both angle values are equal.

Add 5 on both sides.

Divide by 5 on both sides.

The answer is:  

Opposite angles of two intersecting lines must equal to each other. Set up an equation such that both angle are equal.

Add 9 on both sides.

Subtract  on both sides.

This means that  equals .

Since we have two parallel lines, we know that  since they are opposite angle. 

We also know that  are supplementary because they are consecutive interior angles. Thus, we know that  is also supplementary to .

We can then set up the following equation to solve for .

Thus,  and .

Now, notice that  because they are corresponding angles. Thus, .

Start by noticing that the two angles with the values of  and  are supplementary.

Thus, we can write the following equation and solve for .

Since  and  are vertical angles, they must also have the same value.

Thus,