### All Intermediate Geometry Resources

## Example Questions

### Example Question #96 : Triangles

A triangle has two equivalent interior angles, each with a measurement of degrees. Select the most accurate categorization of this triangle.

**Possible Answers:**

Acute isosceles triangle.

Isosceles right triangle.

Obtuse isosceles triangle.

Acute scalene triangle.

Right scalene triangle.

**Correct answer:**

Isosceles right triangle.

The sum of the three interior angles in any triangle must equal degrees. Since this triangle has two equivalent interior angles, it must be an isosceles triangle. Additionally, the sum of the two equivalent interior angles is: .

Since, degrees, the third interior angle must be a right angle--which makes this triangle an isosceles right triangle.

### Example Question #97 : Triangles

A particular triangle does not have any equivalent sides, and has one interior angle with a measurement of degrees.

Select the most accurate categorization of this triangle.

**Possible Answers:**

Acute isosceles triangle.

Obtuse scalene triangle.

Right scalene triangle.

Acute scalene triangle.

Right isosceles triangle.

**Correct answer:**

Obtuse scalene triangle.

A scalene triangle doesn't have any equivalent sides or equivalent interior angles. Since this triangle has one obtuse interior angle, it must be an obtuse scalene triangle.

### Example Question #101 : Triangles

A triangle has interior angles that range from degrees to degrees. What is the most accurate categorization of this triangle?

**Possible Answers:**

Right triangle.

Acute isosceles triangle.

Obtuse scalene triangle.

Obtuse isosceles triangle.

Acute scalene triangle.

**Correct answer:**

Acute scalene triangle.

Acute scalene triangles must have three different acute interior angles--which always have a sum of degrees.

Thus, the solution is:

### Example Question #101 : Triangles

A particular triangle has interior angle measurements that range from degrees to degrees. What is the most accurate categorization of this triangle?

**Possible Answers:**

Right triangle.

Obtuse isosceles triangle.

Acute isosceles triangle.

Acute scalene triangle.

Obtuse scalene triangle.

**Correct answer:**

Obtuse isosceles triangle.

An obtuse isosceles triangle has one obtuse interior angle and two equivalent acute interior angles. Since the sum total of the interior angles of every triangle must equal degrees, the solution is:

Therefore, this triangle has one interior angle of degrees and two equivalent interior angles of degrees--making this an obtuse isosceles triangle.

### Example Question #102 : Triangles

A triangle has sides of lengths 18.4, 18.4, and 26.0. Is the triangle acute, right, or obtuse?

**Possible Answers:**

Right

Obtuse

Acute

**Correct answer:**

Acute

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

Calculate the square of the length of the longest side:

The former is greater than the latter. This indicates that the triangle is acute.

### Example Question #103 : Triangles

A triangle has sides of lengths 19.5, 46.8, and 50.7. Is the triangle acute, right, or obtuse?

**Possible Answers:**

Right

Obtuse

Acute

**Correct answer:**

Right

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

Calculate the square of the length of the longest side:

The two quantities are equal, so by the Converse of the Pythagorean Theorem, the triangle is right.

### Example Question #44 : Acute / Obtuse Triangles

A triangle has sides of lengths 9, 12, and 18. Is the triangle acute, right, or obtuse?

**Possible Answers:**

Obtuse

Right

Acute

**Correct answer:**

Obtuse

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

Calculate the square of the length of the longest side:

The former is less than the latter. This indicates that the triangle is obtuse.

### Example Question #42 : Acute / Obtuse Triangles

A triangle has sides of lengths 14, 18, and 20. Is the triangle acute, right, or obtuse?

**Possible Answers:**

Acute

Right

Obtuse

**Correct answer:**

Acute

Calculate the sum of the squares of the lengths of the two shortest sides:

Calculate the square of the length of the longest side:

The former is greater than the latter. This indicates that the triangle is acute.

### Example Question #104 : Triangles

Two of the interior angles of a triangle have measure and . Is the triangle acute, right, or obtuse?

**Possible Answers:**

Right

Obtuse

Acute

**Correct answer:**

Acute

The measures of the interior angles of a triangle add up to . If is the measure of the third angle, then

Solve for :

Each of the three angles has measure less than , so each angle is, by definition, acute. This makes the triangle acute.

### Example Question #105 : Triangles

Two of the exterior angles of a triangle, taken at *different* vertices, measure . Is the triangle acute, right, or obtuse?

**Possible Answers:**

Right

Acute

Obtuse

**Correct answer:**

Acute

At a given vertex, an exterior angle and an interior angle of a triangle form a linear pair, making them supplementary - that is, their measures total . The measures of two interior angles can be calculated by subtracting the exterior angle measures from :

The triangle has two interior angles of measure .

The measures of the interior angles of a triangle add up to . If is the measure of the third angle, then

All three interior angles measure less than , making them acute. The triangle is, by definition, acute.