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

## Example Questions

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

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

This triangle is isosceles and obtuse.

This triangle is isosceles and right.

This triangle is scalene and obtuse.

This triangle cannot exist.

This triangle is scalene and right.

This triangle cannot exist.

Explanation:

A triangle must have at least two acute angles; however, a triangle with angles that measure  and  could have at most one acute angle, an impossible situation. Therefore, this triangle is nonexistent.

### Example Question #5 : Acute / Obtuse Triangles

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

The triangle cannot exist.

The triangle is acute and isosceles.

The triangle is obtuse and isosceles.

The triangle is obtuse and scalene.

The triangle is acute and scalene.

The triangle cannot exist.

Explanation:

A triangle must have at least two acute angles; however, a triangle with angles that measure  would have two obtuse angles and at most one acute angle. This is not possible, so this triangle cannot exist.

### Example Question #5 : Acute / Obtuse Triangles

One angle of an isosceles triangle has measure . What are the measures of the other two angles?

Not enough information is given to answer this question.

Explanation:

An isosceles triangle not only has two sides of equal measure, it has two angles of equal measure. This means one of two things, which we examine separately:

Case 1: It has another  angle. This is impossible, since a triangle cannot have two obtuse angles.

Case 2: Its other two angles are the ones that are of equal measure. If we let  be their common measure, then, since the sum of the measures of a triangle is

Both angles measure

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

The angles of a triangle measure . Evaluate

Explanation:

The sum of the degree measures of the angles of a triangle is 180, so we solve for  in the following equation:

### Example Question #6 : Acute / Obtuse Triangles

The acute angles of a right triangle measure  and

Evaluate .

Explanation:

The degree measures of the acute angles of a right triangle total 90, so we solve for  in the following equation:

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

Note: Figure NOT drawn to scale

Refer to the above figure. .

What is the measure of  ?

Explanation:

Congruent chords of a circle have congruent minor arcs, so since , and their common measure is .

Since there are  in a circle,

The inscribed angle  intercepts this arc and therefore has one-half its degree measure, which is

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

Solve for :

Explanation:

The sum of the internal angles of a triangle is equal to . Therefore:

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

Refer to the above figure. Express  in terms of .

Explanation:

The measure of an interior angle of a triangle is equal to 180 degrees minus that of its adjacent exterior angle, so

and

.

The sum of the degree measures of the three interior angles is 180, so

### Example Question #92 : Plane Geometry

In the above figure, .

Give the measure of .

Explanation:

and  form a linear pair, so their degree measures total ; consequently,

, so by the Isosceles Triangle Theorem,

The sum of the degree measures of a triangle is , so

### Example Question #93 : Plane Geometry

Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Explanation:

The measure of an exterior angle of a triangle, which here is , is equal to the sum of the measures of its remote interior angles, which here are  and . Consequently,

and  form a linear pair and, therefore,

.

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