### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Lines

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 #81 : Geometry

Which is the greater quantity?

(a) The measure of an angle complementary to a angle

(b) The measure of an angle supplementary to a angle

**Possible Answers:**

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

**Correct answer:**

(b) is greater

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

(a) The measure of an angle complementary to a angle is

(b) The measure of an angle supplementary to a angle is

This makes (b) greater.

### Example Question #2 : Lines

and are complementary; .

Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) and (B) are equal

(A) is greater

It is impossible to determine which is greater from the information given

(B) is greater

**Correct answer:**

(B) is greater

Two angles are complementary if their degree measures total 90. Therefore,

Since , we can substitute, and we can solve for :

, making (B) the greater quantity.

### Example Question #3 : Lines

Note: diagram is *not* drawn to scale

Refer to the above diagram. If , what is ?

**Possible Answers:**

**Correct answer:**

and form a linear pair, so

Since , this can be rewritten as

, and the first equation can be rewritten as:

### Example Question #81 : Plane Geometry

Note: Figure NOT drawn to scale.

Which of the following pairs of numbers could give the measures of and ?

**Possible Answers:**

None of the pairs given in the other choices is correct.

**Correct answer:**

The two angles form a linear pair and therefore their measures total . We check all of the pairs for this sum.

The correct pair is .

### Example Question #3 : Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which of the following triples could refer to the measures of , , and ?

**Possible Answers:**

All of the other responses are correct.

**Correct answer:**

All of the other responses are correct.

The measure of an exterior angle of a triangle, which here is , is the sum of the measures of its remote interior angles, which here are and . Therefore, we are looking for the sum of the first two angle measures to be equal to the third.

All four triples satisfy this condition.

### Example Question #4 : Lines

Note: figure NOT drawn to scale

The degree measure of is five degrees greater than twice that of . Which is the greater quantity?

(A)

(B)

**Possible Answers:**

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(A) is greater

(B) is greater

**Correct answer:**

(A) is greater

The degree measure of is five degrees greater than twice that of - that is,

Since and form a linear pair,

, and by substitution,

This makes (A) greater.

### Example Question #6 : Lines

Note: Figure NOT drawn to scale

. What is ?

**Possible Answers:**

**Correct answer:**

and form a linear pair and therfore, the the sum of their degree measures is .

### Example Question #1 : Lines

Note: Figure NOT drawn to scale.

Evaluate .

**Possible Answers:**

**Correct answer:**

, so

The measure of an exterior angle of a triangle, which here is , is the sum of the measures of its remote interior angles, which here are and .

### Example Question #1 : Lines

What is the slope of a line that passes through points and ?

**Possible Answers:**

**Correct answer:**

The equation for solving for the slope of a line is .

Thus, if and , then: