SSAT Upper Level Math : How to find an angle in an acute / obtuse triangle

Study concepts, example questions & explanations for SSAT Upper Level Math

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Example Questions

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Triangle_a

Figure NOT drawn to scale.

If  and , evaluate .

Possible Answers:

Correct answer:

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

If the vertex angle of an isoceles triangle is , what is the value of one of its base angles?

Possible Answers:

Correct answer:

Explanation:

In an isosceles triangle, the base angles are the same. Also, the three angles of a triangle add up to .

So, subtract the vertex angle from . You get .

Because there are two base angles you divide  by , and you get .

Example Question #3 : How To Find An Angle In An Acute / Obtuse Triangle

Triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. 

Which of the following could be a measure of  ?

Possible Answers:

All of the other choices give a possible measure of .

Correct answer:

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

.

We also have the following constraints:

Then, by the addition property of inequalities,

Therefore, the measure of  must fall in that range. Of the given choices, only  falls in that range.

Example Question #4 : How To Find An Angle In An Acute / Obtuse Triangle

Triangle

Refer to the above diagram. 

Which of the following could be a measure of  ?

Possible Answers:

All of the other responses are correct.

Correct answer:

All of the other responses are correct.

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

or 

Therefore, the maximum value of  is the least possible value of  subtracted from the greatest possible value of :

The minimum value of  is the greatest possible value of  subtracted from the least possible value of :

Therefore, 

Since all of the choices fall in this range, all are possible measures of .

Example Question #5 : How To Find An Angle In An Acute / Obtuse Triangle

Find the angle measurement of .

 

Picture1

Possible Answers:

Correct answer:

Explanation:

All the angles in a triangle must add up to .

Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Find the angle measurement of .

 

 

Picture2

Possible Answers:

Correct answer:

Explanation:

All the angles in a triangle must add up to 

Example Question #7 : How To Find An Angle In An Acute / Obtuse Triangle

Find the angle measurement of .

 

 

Picture3

Possible Answers:

Correct answer:

Explanation:

All the angles in a triangle must add up to .

Example Question #6 : How To Find An Angle In An Acute / Obtuse Triangle

An isosceles triangle has an angle whose measure is .

What could be the measures of one of its other angles?

(a) 

(b)  

(c) 

Possible Answers:

(a) or (c) only

(a), (b), or (c)

(b) only

(c) only

(a) only

Correct answer:

(a), (b), or (c)

Explanation:

By the Isosceles Triangle Theorem, an isosceles triangle has two congruent interior angles. There are two possible scenarios if one angle has measure :

Scenario 1: The other two angles are congruent to each other. The degree measures of the interior angles of a triangle total , so if we let  be the common measure of those angles:

This makes (b) a possible answer.

Scenario 2: One of the other angles measures  also, making (c) a possible answer. The degree measure of the third angle is

,

making (a) a possible answer. Therefore, the correct choice is (a), (b), or (c).

Example Question #8 : How To Find An Angle In An Acute / Obtuse Triangle

One of the interior angles of a scalene triangle measures . Which of the following could be the measure of another of its interior angles?

Possible Answers:

Correct answer:

Explanation:

A scalene triangle has three sides of different measure, so, by way of the Converse of the Isosceles Triangle Theorem, each angle is of different measure as well. We can therefore eliminate  immediately. 

Also, if the triangle also has a  angle, then, since the total of the degree measures of the angles is , it follows that the third angle has measure

.

Therefore, the triangle has two angles that measure the same, and  can be eliminated.

Similarly, if the triangle also has a  angle, then, since the total of the degree measures of the angles is , it follows that the third angle has measure

.

The triangle has two angles that measure . This choice can be eliminated.

 can be eliminated, since the third angle would have measure

,

an impossible situation since angle measures must be positive.

The remaining possibility is . This would mean that the third angle has measure

.

The three angles have different measures, so the triangle is scalene.  is the correct choice.

Example Question #7 : How To Find An Angle In An Acute / Obtuse Triangle

Given:  with . Locate  on  so that  is the angle bisector of . What is  ?

Possible Answers:

Correct answer:

Explanation:

Angle bisector

Above is the figure described.

The measures of the interior angles of a triangle total , so the measure of  is

Since  bisects this angle, 

and 

 

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