# GED Math : Opposite and Corresponding Angles

## Example Questions

### Example Question #1 : Opposite And Corresponding Angles

Refer to the above diagram.

Which of the following is a valid alternative name for  ?

Explanation:

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 .

### Example Question #2 : Opposite And Corresponding Angles

Refer to the above diagram.

.

What is  ?

Explanation:

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

### Example Question #3 : Opposite And Corresponding Angles

Refer to the above diagram.

Which of the following facts does not, by itself, prove that  ?

and  are supplementary angles

bisects

is a right angle

bisects

Explanation:

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.

### Example Question #4 : Opposite And Corresponding Angles

In two intersecting lines, the opposite angles are  and .  What must be the value of ?

Explanation:

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.

Divide by 7 on both sides.

### Example Question #5 : Opposite And Corresponding Angles

Suppose a pair of opposite angles are measured  and .  What must the value of ?

Explanation:

Vertical angles are equal.

Set both angles equal and solve for x.

Subtract  on both sides.

Divide by 4 on both sides.

### Example Question #6 : Opposite And Corresponding Angles

Suppose two vertical angles in a pair of intersecting lines.  What is the value of  if one angle is  and the other angle is ?

Explanation:

Vertical angles of intersecting lines must equal to each other.

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

Divide by three on both sides.

### Example Question #7 : Opposite And Corresponding Angles

Suppose two opposite angles are measured  and . What is the value of ?

Explanation:

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

Divide by 5 on both sides.

### Example Question #8 : Opposite And Corresponding Angles

With a pair of intersecting lines, a set of opposite angles are measured  and .  What must the value of  be?

Explanation:

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

Subtract  on both sides.

This means that  equals .

### Example Question #9 : Opposite And Corresponding Angles

In the figure above, . If the measure of  and , what is the measure of ?

Explanation:

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, .

### Example Question #10 : Opposite And Corresponding Angles

Find the value of .

Assume the two horizontal lines are parallel.

Explanation:

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,