### All ISEE Upper Level Math Resources

## Example Questions

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

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

**Possible Answers:**

**Correct answer:**

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 #1 : How To Find An Angle

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

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

By angle addition,

### Example Question #1 : Lines

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

**Possible Answers:**

**Correct answer:**

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 #2 : How To Find An Angle

and are supplementary; and are complementary.

.

What is ?

**Possible Answers:**

**Correct answer:**

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

, so its supplement has measure

, the complement of , has measure

### Example Question #2 : Lines

Note: Figure NOT drawn to scale.

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

**Possible Answers:**

**Correct answer:**

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

### Example Question #1 : 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?

**Possible Answers:**

**Correct answer:**

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 #1 : 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?

**Possible Answers:**

**Correct answer:**

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 #3 : Lines

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

Evaluate .

**Possible Answers:**

**Correct answer:**

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 #1 : How To Find An Angle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the measure of .

**Possible Answers:**

**Correct answer:**

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, .