### All Intermediate Geometry Resources

## Example Questions

### Example Question #51 : Acute / Obtuse Triangles

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

**Possible Answers:**

Right

Acute

Obtuse

**Correct answer:**

Right

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 measures and . The sum of these measures is , thereby making them complementary. A triangle with two complementary acute angles is a right triangle.

### Example Question #51 : Acute / Obtuse Triangles

True or false: It is possible for a triangle to have angles of measure , , and .

**Possible Answers:**

False

True

**Correct answer:**

True

The sum of the measures of the angles of a triangle is . The sum of the three given angle measures is

.

This makes the triangle possible.

### Example Question #552 : Intermediate Geometry

True or false: It is possible for a triangle to have three interior angles, each of whose measures are .

**Possible Answers:**

False

True

**Correct answer:**

False

A triangle with three congruent angles is an equiangular - and equilateral - triangle; such an angle must have three angles that measure .

### Example Question #54 : Acute / Obtuse Triangles

Given: with perimeter 40;

True or false:

**Possible Answers:**

True

False

**Correct answer:**

True

The perimeter of is the sum of the lengths of its sides - that is,

The perimeter is 40, so set , and solve for :

Subtract 26 from both sides:

, so by the Isosceles Triangle Theorem, their opposite angles are congruent - that is,

.

### Example Question #22 : How To Find An Angle In An Acute / Obtuse Triangle

is an equilateral triangle; is the midpoint of ; the segment is constructed.

True or false: .

**Possible Answers:**

False

True

**Correct answer:**

False

The referenced triangle is below:

In an equilateral triangle, the median from - the segment from to , the midpoint of the opposite side - is also the bisector of the angle , so

Each interior angle of an equilateral triangle, including , measures , so substitute and evaluate:

.

### Example Question #56 : Acute / Obtuse Triangles

is an equilateral triangle. Locate a point along and construct . .

Evaluate .

**Possible Answers:**

**Correct answer:**

The referenced figure is below. Note that , as is the case with all of the interior angles of an equilateral triangle.

The interior angles of an equilateral triangle each measure . An exterior angle of a triangle has as its degree measure the sum of its remote interior angles; specifically,

Substitute the known angle measures, and solve: