### All ISEE Upper Level Math Resources

## Example Questions

### 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 ?

**Possible Answers:**

This triangle is scalene and obtuse.

This triangle is scalene and right.

This triangle is isosceles and right.

This triangle cannot exist.

This triangle is isosceles and obtuse.

**Correct answer:**

This triangle cannot exist.

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

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

**Possible Answers:**

The triangle is obtuse and scalene.

The triangle is acute and isosceles.

The triangle cannot exist.

The triangle is obtuse and isosceles.

The triangle is acute and scalene.

**Correct answer:**

The triangle cannot exist.

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 #1 : Acute / Obtuse Triangles

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

**Possible Answers:**

Not enough information is given to answer this question.

**Correct answer:**

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

The angles of a triangle measure . Evaluate .

**Possible Answers:**

**Correct answer:**

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

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

The acute angles of a right triangle measure and .

Evaluate .

**Possible Answers:**

**Correct answer:**

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

### Example Question #6 : 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 ?

**Possible Answers:**

**Correct answer:**

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

Solve for :

**Possible Answers:**

**Correct answer:**

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

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

Refer to the above figure. Express in terms of .

**Possible Answers:**

**Correct answer:**

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

In the above figure, .

Give the measure of .

**Possible Answers:**

**Correct answer:**

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

Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

**Possible Answers:**

**Correct answer:**

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,

.