### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #11 : Plane Geometry

Refer to the above figure. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

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

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(a) is the greater quantity

Extend as seen in the figure below:

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles; specifically,

,

and

However, , so, by substitution,

### Example Question #12 : Acute / Obtuse Triangles

Given: . . Which is the greater quantity?

(a)

(b)

**Possible Answers:**

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

Below is the referenced triangle along with , an equilateral triangle with sides of length 10:

As an angle of an equilateral triangle, has measure . Applying the Side-Side-Side Inequality Theorem, since , , and , it follows that , so .

Also, since , by the Isosceles Triangle Theorem, . Since , and the sum of the measures of the angles of a triangle is , it follows that

Substituting and solving:

.

### Example Question #13 : Acute / Obtuse Triangles

Given and with

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

**Correct answer:**

It is impossible to tell from the information given.

Examine the diagram below, in which two triangles matching the given descriptions have been superimposed.

Note that and . The two question marks need to be replaced by and . No matter how you place these two points, . However, with one replacement, ; with the other replacement, . Therefore, the information given is insufficient to answer the question.

### Example Question #1 : How To Find If Triangles Are Congruent

Consider and with .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(a) and (b) are equal.

, so, by the Side-Side-Side Principle, since there are three pairs of congruent corresponding sides between the triangles, we can say they are congruent - that is,

.

Corresponding angles of congruent sides are congruent, so .

### Example Question #11 : Geometry

Which of the following could be the lengths of the three sides of a scalene triangle?

**Possible Answers:**

All of the other choices are possible lengths of a scalene triangle

**Correct answer:**

All of the other choices are possible lengths of a scalene triangle

A scalene triangle, by definition, has sides all of different lengths. Since all of the given choices fit that criterion, the correct choice is that all can be scalene.

### Example Question #16 : Acute / Obtuse Triangles

Given with right angle , .

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

**Correct answer:**

(a) is greater

The sum of the measures of the angles of a triangle is 180, so

, so the side opposite , which is , is longer than the side opposite , which is . This makes (a) the greater quantity.

### Example Question #11 : Triangles

Given with obtuse angle , which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(b) is greater.

To compare the lengths of and from the angle measures, it is necessary to know which of their opposite angles - and , respectively - is the greater angle. Since is the obtuse angle, it has the greater measure, and is the longer side. This makes (b) greater.

### Example Question #1 : How To Find The Length Of The Side Of A Triangle

has obtuse angle ; . Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

**Correct answer:**

(a) is greater.

Since is the obtuse angle of ,

.

,

,

so (a) is the greater quantity.

### Example Question #12 : Geometry

Given with . Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

**Correct answer:**

(b) is greater.

Use the Triangle Inequality:

This makes (b) the greater quantity.

### Example Question #11 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Given with . Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

**Correct answer:**

It is impossible to tell from the information given.

By the Converse of the Pythagorean Theorem,

if and only if is a right angle.

However, if is acute, then ; if is obtuse, then .

Since we do not know whether is acute, right, or obtuse, we cannot determine whether (a) or (b) is greater.

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