GMAT Math : Acute / Obtuse Triangles

Example Questions

Example Question #11 : Acute / Obtuse Triangles

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

Explanation:

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  ?

Explanation:

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  ?

is isosceles and obtuse.

may be scalene or isosceles, but it is obtuse.

may be scalene or isosceles, but it is acute,

is scalene and acute.

is scalene and obtuse.

is scalene and obtuse.

Explanation:

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 ?

The triangle must be right and isosceles.

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

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

The triangle cannot exist.

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

The triangle cannot exist.

Explanation:

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?

Explanation:

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

Explanation:

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 #11 : Acute / Obtuse Triangles

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

is acute and isosceles

is acute and scalene

is obtuse and scalene

is right and scalene

is obtuse and isosceles

is acute and scalene

Explanation:

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 #92 : Triangles

Note: Figure NOT drawn to scale.

Refer to the above diagram.

Evaluate .

Explanation:

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

Example Question #93 : Triangles

In the following triangle:

The angle  degrees

The angle  degrees

(Figure not drawn on scale)

Find the value of .

Explanation:

Since , the following triangles are isoscele: .

If ADC, BDC, and BDA are all isoscele; then:

The angle  degrees

The angle  degrees, and

The angle  degrees

Therefore:

The angle

The angle  degrees, and

The angle

Since the sum of angles of a triangle is equal to 180 degrees then:

. So:

.

Now let us solve the equation for x:

(See image below - not drawn on scale)

Example Question #94 : Triangles

Which of the following is true of a triangle with two  angles?

The triangle must be obtuse but it can be either scalene or isosceles.

The triangle must be isosceles and acute.

The triangle must be isosceles but it can be acute, right, or obtuse.

The triangle must be isosceles and obtuse.

The triangle must be scalene and obtuse.