ISEE Upper Level Quantitative : Acute / Obtuse Triangles

Example Questions

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

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

80 degrees

65 degrees

95  degrees

50 degrees

95  degrees

Explanation:

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

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

Explanation:

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 #3 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Let the three angles of a triangle measure , and .

Which of the following expressions is equal to  ?

Explanation:

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

Example Question #4 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

The triangle is acute and scalene.

The triangle is obtuse and isosceles.

The triangle is obtuse and scalene.

The triangle cannot exist.

The triangle is acute and isosceles.

The triangle is obtuse and isosceles.

Explanation:

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 #5 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

You are given two triangles,  and .

is an acute angle, and  is a right angle.

Which quantity is greater?

(a)

(b)

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

Explanation:

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 #6 : 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)

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) is greater.

Explanation:

(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 #7 : 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)

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) and (b) are equal.

Explanation:

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 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

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

(a)

(b)

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

Explanation:

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 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

is equilateral;  is isosceles

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

(a) is greater.

Explanation:

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 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which is the greater quantity?

(a)

(b)

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

(a) and (b) are equal

Explanation:

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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