# ISEE Upper Level Quantitative : Geometry

## Example Questions

### Example Question #11 : Plane Geometry

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

(a)

(b)

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

(a) is the greater quantity

Explanation:

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)

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

(b) is the greater quantity

Explanation:

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)

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

It is impossible to tell from the information given.

Explanation:

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)

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

(a) and (b) are equal.

Explanation:

, 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?

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

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

Explanation:

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)

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

(a) is greater

Explanation:

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)

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Explanation:

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)

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) is greater.

Explanation:

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)

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Explanation:

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)

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.