# ISEE Upper Level Math : How to find an angle

## Example Questions

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### Example Question #2 : Lines

Examine the above diagram. If , give  in terms of .

Explanation:

The two marked angles are same-side exterior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for  in this equation:

### Example Question #3 : Lines

Examine the above diagram. If , give  in terms of .

Explanation:

The two marked angles are same-side interior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for  in this equation:

### Example Question #1 : How To Find An Angle

Examine the above diagram. What is  ?

Explanation:

### Example Question #5 : Lines

Examine the above diagram. Which of the following statements must be true whether or not  and  are parallel?

Explanation:

Four statements can be eliminated by the various parallel theorems and postulates. Congruence of alternate interior angles or corresponding angles forces the lines to be parallel, so

and

.

Also, if same-side interior angles or same-side exterior angles are supplementary, the lines are parallel, so

and

.

However,  whether or not  since they are vertical angles, which are always congruent.

### Example Question #6 : Lines

and  are supplementary;  and  are complementary.

.

What is  ?

Explanation:

Supplementary angles and complementary angles have measures totaling  and , respectively.

, so its supplement  has measure

, the complement of , has measure

### Example Question #2 : How To Find An Angle

Note: Figure NOT drawn to scale.

In the above figure,  and . Which of the following is equal to  ?

Explanation:

and  form a linear pair, so their angle measures total . Set up and solve the following equation:

### Example Question #3 : How To Find An Angle

Two angles which form a linear pair have measures  and . Which is the lesser of the measures (or the common measure) of the two angles?

Explanation:

Two angles that form a linear pair are supplementary - that is, they have measures that total . Therefore, we set and solve for  in this equation:

The two angles have measure

and

is the lesser of the two measures and is the correct choice.

### Example Question #4 : How To Find An Angle

Two vertical angles have measures  and . Which is the lesser of the measures (or the common measure) of the two angles?

Explanation:

Two vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure. Therefore, we set up and solve the equation

### Example Question #41 : Plane Geometry

A line  intersects parallel lines  and  and  are corresponding angles;  and  are same side interior angles.

Evaluate .

Explanation:

When a transversal such as  crosses two parallel lines, two corresponding angles - angles in the same relative position to their respective lines - are congruent. Therefore,

Two same-side interior angles are supplementary - that is, their angle measures total 180 - so

We can solve this system by the substitution method as follows:

Backsolve:

, which is the correct response.

### Example Question #5 : How To Find An Angle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the measure of .

Explanation:

The top and bottom angles, being vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure, so

or, simplified,

The right and bottom angles form a linear pair, so their degree measures total 180. That is,

Substitute  for :

The left and right angles, being vertical angles, have the same measure, so, since the right angle measures , this is also the measure of the left angle, .

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