### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Plane Geometry

In isosceles triangle *ABC*, the measure of angle *A* is 50 degrees. Which is NOT a possible measure for angle *B*?

**Possible Answers:**

65 degrees

80 degrees

95 degrees

50 degrees

**Correct answer:**

95 degrees

If angle *A* is one of the base angles, then the other base angle must measure 50 degrees. Since 50 + 50 + *x* = 180 means *x* = 80, the vertex angle must measure 80 degrees.

If angle *A* is the vertex angle, the two base angles must be equal. Since 50 + *x* + *x* = 180 means *x* = 65, the two base angles must measure 65 degrees.

The only number given that is not possible is 95 degrees.

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

The angles of a triangle measure , , and . Give in terms of .

**Possible Answers:**

**Correct answer:**

The sum of the measures of three angles of a triangle is , so we can set up the equation:

We can simplify and solve for :

### Example Question #1 : Geometry

Let the three angles of a triangle measure , , and .

Which of the following expressions is equal to ?

**Possible Answers:**

**Correct answer:**

The sum of the measures of the angles of a triangle is , so simplify and solve for in the equation:

### Example Question #2 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which of the following is true about a triangle with two angles that measure each?

**Possible Answers:**

The triangle cannot exist.

The triangle is acute and scalene.

The triangle is obtuse and isosceles.

The triangle is obtuse and scalene.

The triangle is acute and isosceles.

**Correct answer:**

The triangle is obtuse and isosceles.

The measures of the angles of a triangle total , so if two angles measure and we call the measure of the third, then

This makes the triangle obtuse.

Also, since the triangle has two congruent angles (the angles), the triangle is also isosceles.

### Example Question #1 : Acute / Obtuse Triangles

You are given two triangles, and .

, is an acute angle, and is a right angle.

Which quantity is greater?

(a)

(b)

**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

We invoke the SAS Inequality Theorem, which states that, given two triangles and , with , ( the included angles), then - that is, the side opposite the greater angle has the greater length. Since is an acute angle, and is a right angle, we have just this situation. This makes (b) the greater.

### Example Question #1 : Acute / Obtuse Triangles

Note: Figure NOT drawn to scale.

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

(a)

(b)

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(a) is greater.

(a) The measures of the angles of a linear pair total 180, so:

(b) The Triangle Exterior-Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. Therefore, .

Therefore (a) is the greater quantity.

### Example Question #1 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Note: Figure NOT drawn to scale.

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

(a)

(b)

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

**Correct answer:**

(a) and (b) are equal.

The two angles at bottom are marked as congruent. Each of these two angles forms a linear pair with a angle, so it is supplementary to that angle, making its measure . Therefore, the other marked angle also measures .

The sum of the measures of the interior angles of a triangle is , so:

The quantities are equal.

### Example Question #8 : Acute / Obtuse Triangles

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

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

**Correct answer:**

(a) and (b) are equal.

The Triangle Exterior-Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. Therefore,

,

making the quantities equal.

### Example Question #9 : Acute / Obtuse Triangles

is equilateral; is isosceles

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:**

(a) is greater.

is equilateral, so

.

In , we are given that

.

Since the triangles have two pair of congruent sides, the third side with the greater length is opposite the angle of greater measure. Therefore,

.

Since is an angle of an equilateral triangle, its measure is , so .

### Example Question #10 : Acute / Obtuse Triangles

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) and (b) are equal

(b) is the greater quantity

**Correct answer:**

(a) and (b) are equal

Corresponding angles of similar triangles are congruent, so, since , it follows that

By similarity, and , and we are given that , so

Also,

,

and .

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