ISEE Upper Level Quantitative : How to find an angle

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

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Example Question #1 : How To Find An Angle

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

Possible Answers:

50 degrees

80 degrees

95  degrees

65 degrees

Correct answer:

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 #1 : How To Find An Angle

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

Possible Answers:

Correct answer:

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 : How To Find An Angle

Let the three angles of a triangle measure , and .

Which of the following expressions is equal to  ?

Possible Answers:

Correct answer:

Explanation:

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

Example Question #1 : Acute / Obtuse Triangles

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

Possible Answers:

The triangle is acute and isosceles.

The triangle is acute and scalene.

The triangle cannot exist.

The triangle is obtuse and scalene.

The triangle is obtuse and isosceles.

Correct answer:

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 : How To Find An Angle

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

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 #3 : Geometry

Right_triangle

Note: Figure NOT drawn to scale.

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

(a) 

(b) 

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(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 #2 : Acute / Obtuse Triangles

Exterior_angle

Note: Figure NOT drawn to scale.

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

(a) 

(b)

Possible Answers:

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Correct answer:

(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 : How To Find An Angle

Exterior_angle

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

(a)

(b) 

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Correct answer:

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

 is equilateral;  is isosceles

 

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.

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 : How To Find An Angle

 

 

Which is the greater quantity?

(a) 

(b) 

 

Possible Answers:

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

Correct answer:

(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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