### All GMAT Math Resources

## Example Questions

### Example Question #1 : Calculating An Angle In An Acute / Obtuse Triangle

Which of the following cannot be the measure of a base angle of an isosceles triangle?

**Possible Answers:**

Each of the other choices can be the measure of a base angle of an isosceles triangle.

**Correct answer:**

An isosceles triangle has two congruent angles by the Isosceles Triangle Theorem; these are the base angles. Since at least two angles of any triangle must be acute, a base angle must be acute - that is, it must measure under . The only choice that does not fit this criterion is , making this the correct choice.

### Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

Let the three interior angles of a triangle measure , and . Which of the following statements is true about the triangle?

**Possible Answers:**

The triangle is scalene and obtuse.

The triangle is isosceles and acute.

The triangle is scalene and acute.

The triangle is scalene and right.

The triangle is isosceles and obtuse.

**Correct answer:**

The triangle is isosceles and acute.

If these are the measures of the interior angles of a triangle, then they total . Add the expressions, and solve for .

One angle measures . The others measure:

All three angles measure less than , so the triangle is acute. Also, there are two congruent angles, so by the converse of the Isosceles Triangle Theorem, two sides are congruent, and the triangle is isosceles.

### Example Question #11 : Acute / Obtuse Triangles

Two angles of an isosceles triangle measure and . What are the possible values of ?

**Possible Answers:**

**Correct answer:**

In an isosceles triangle, at least two angles measure the same. Therefore, one of three things happens:

Case 1: The two given angles have the same measure.

The angle measures are , making the triangle equianglular and, subsequently, equilateral. An equilateral triangle is considered isosceles, so this is a possible scenario.

Case 2: The third angle has measure .

Then, since the sum of the angle measures is 180,

as before

Case 3: The third angle has measure

as before.

Thus, the only possible value of is 40.

### Example Question #12 : Acute / Obtuse Triangles

Two angles of an isosceles triangle measure and . What are the possible value(s) of ?

**Possible Answers:**

**Correct answer:**

In an isosceles triangle, at least two angles measure the same. Therefore, one of three things happens:

Case 1: The two given angles have the same measure.

This is a false statement, indicating that this situation is impossible.

Case 2: The third angle has measure .

Then, since the sum of the angle measures is 180,

This makes the angle measures , a plausible scenario.

Case 3: the third angle has measure

Then, since the sum of the angle measures is 180,

This makes the angle measures , a plausible scenario.

Therefore, either or

### Example Question #13 : Acute / Obtuse Triangles

Which of the following is true of ?

**Possible Answers:**

is scalene and acute.

may be scalene or isosceles, but it is acute,

may be scalene or isosceles, but it is obtuse.

is scalene and obtuse.

is isosceles and obtuse.

**Correct answer:**

is scalene and obtuse.

By similarity, .

Since measures of the interior angles of a triangle total ,

Since the three angle measures of are all different, no two sides measure the same; the triangle is scalene. Also, since, the angle is obtuse, and is an obtuse triangle.

### Example Question #14 : Acute / Obtuse Triangles

Which of the following is true of a triangle with three angles whose measures have an arithmetic mean of ?

**Possible Answers:**

The triangle must be right and isosceles.

The triangle must be obtuse but may be scalene or isosceles.

The triangle may be right or obtuse but must be scalene.

The triangle must be right but may be scalene or isosceles.

The triangle cannot exist.

**Correct answer:**

The triangle cannot exist.

The sum of the measures of three angles of any triangle is 180; therefore, their mean is , making a triangle with angles whose measures have mean 90 impossible.

### Example Question #15 : Acute / Obtuse Triangles

Two angles of a triangle measure and . What is the measure of the third angle?

**Possible Answers:**

**Correct answer:**

The sum of the degree measures of the angles of a triangle is 180, so we can subtract the two angle measures from 180 to get the third:

### Example Question #16 : Acute / Obtuse Triangles

The angles of a triangle measure . Evaluate .

**Possible Answers:**

**Correct answer:**

The sum of the measures of the angles of a triangle total , so we can set up and solve for in the following equation:

### Example Question #17 : Acute / Obtuse Triangles

An exterior angle of with vertex measures ; an exterior angle of with vertex measures . Which is the following is true of ?

**Possible Answers:**

is obtuse and isosceles

is acute and isosceles

is obtuse and scalene

is right and scalene

is acute and scalene

**Correct answer:**

is acute and scalene

An interior angle of a triangle measures minus the degree measure of its exterior angle. Therefore:

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

.

Each angle is acute, so the triangle is acute; each angle is of a different measure, so the triangle has three sides of different measure, making it scalene.

### Example Question #18 : Acute / Obtuse Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram.

Evaluate .

**Possible Answers:**

**Correct answer:**

The sum of the exterior angles of a triangle, one per vertex, is . , and are exterior angles at different vertices, so

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