# GMAT Math : Acute / Obtuse Triangles

## Example Questions

### Example Question #95 : Triangles

The measures of the interior angles of a triangle are , and . Also,

.

Evaluate .

Possible Answers:

Correct answer:

Explanation:

The measures of the interior angles of a triangle have sum , so

Along with , a system of linear equations is formed that can be solved by adding:

### Example Question #96 : Triangles

The interior angles of a triangle have measures , and . Also,

.

Which of the following is closest to ?

Possible Answers:

Correct answer:

Explanation:

The measures of the interior angles of a triangle have sum , so

, or

Along with , a system of linear equations is formed that can be solved by adding:

Of the given choices, 50 comes closest to the correct measure.

### Example Question #97 : Triangles

A triangle has interior angles whose measures are  , and . A second triangle has interior angles, two of whose measures are  and . What is the measure of the third interior angle of the second triangle?

Possible Answers:

None of the other responses gives the correct answer.

Correct answer:

Explanation:

The measures of the interior angles of a triangle have sum , so

, or, equivalently,

and  are the measures of two interior angles of the second triangle, so if we let  be the measure of the third angle, then

By substitution,

and

.

The correct response is .

### Example Question #98 : Triangles

The measures of the interior angles of Triangle 1 are   , and . The measures of two of the interior angles of Triangle 2 are  and . Which of the following is the measure of the third interior angle of Triangle 2?

Possible Answers:

Correct answer:

Explanation:

The measures of the interior angles of a triangle have sum , so

, or, equivalently,

and  are the measures of two interior angles of the second triangle, so if we let  be the measure of the third angle, then

By substitution,

The correct response is .

### Example Question #99 : Triangles

Triangle 1 has three interior angles with measures , and . Triangle 1 has three interior angles with measures , and

Express  in terms of .

Possible Answers:

Correct answer:

Explanation:

The sum of the measures of the interior angles of a triangle is , so it can be determined from Triangle 1 that

From Triangle 2, we can deduce that

By substitution:

### Example Question #100 : Triangles

Is  an acute triangle, a right triangle, or an obtuse triangle?

Statement 1:

Statement 2:

Possible Answers:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. The sum of the measures of interior angles of a triangle is ;

, or, equivalently, for some positive number

,

so

Therefore, , making  obtuse, and  an obtuse triangle.

Assume Statement 2 alone. Since the sum of the squares of the lengths of two sides exceeds the square of the length of the third, it follows that  is an obtuse triangle.

### Example Question #21 : Acute / Obtuse Triangles

is an exterior angle of  at .

Is  an acute triangle, a right triangle, or an obtuse triangle?

Statement 1:  is acute.

Statement 2:  and  are both acute.

Possible Answers:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Exterior angle  forms a linear pair with its interior angle . Either both are right, or one is acute and one is obtuse. From Statement 1 alone, since  is acute,  is obtuse, and  is an obtuse triangle.

Statement 2 alone is insufficient. Every triangle has at least two acute angles, and Statement 2 only establishes that   and  are both acute; the third angle, , can be acute, right, or obtuse, so the question of whether  is an acute, right, or obtuse triangle is not settled.

### Example Question #351 : Geometry

, and  are all exterior angles of  with vertices , and , respectively.

Is  an acute triangle, a right triangle, or an obtuse triangle?

Statement 1: , and  are all obtuse angles.

Statement 2: .

Note: For purposes of this problem, , , and  will refer to the interior angles of the triangle at these vertices.

Possible Answers:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. An exterior angle of a triangle forms a linear pair with the interior angle of the triangle of the same vertex. The two angles, whose measures total , must be two right angles or one acute angle and one obtuse angle. Since , and  are all obtuse angles, it follows that their respective interior angles - the three angles of  - are all acute. This makes  an acute triangle.

Statement 2 alone provides insufficient information to answer the question. For example, if  and  each measure  and  measures , the sum of the angle measures is  and  are congruent, and  is an obtuse angle (measuring more than ); this makes  an obtuse triangle. But  if , and  each measure , the sum of the angle measures is again  and  are again congruent, and all three angles are acute (measuring less than ); this makes  an acute triangle.

### Example Question #352 : Geometry

The measures of the angles of one triangle, in degrees, are .

The measures of the angles of a second triangle, in degrees, are .

What is ?

Possible Answers:

Correct answer:

Explanation:

The degree measures of the angles of a triangle add up to a total of 180, so we can set up the following equations:

From the first triangle:

From the second:

These equations form a system of equations that can be solved:

, so

and .

### Example Question #353 : Geometry

The largest angle of an obtuse isosceles triangle has a measure of  .  If the length of the two equivalent sides is  ,  what is the length of the hypotenuse?

Possible Answers:

Correct answer:

Explanation:

The height of the obtuse isosceles triangle bisects the    angle and forms two congruent right triangles. The hypotenuse of each of these triangles is either side of equivalent length, and we can see that the base of either triangle makes up half of the hypotenuse of the obtuse isosceles triangle. Because we know the angle opposite each base is half of  ,  or  ,  we can use the sine of this angle to find the length of the base. As there are two congruent right triangles that make up the obtuse isosceles triangle, the length of either base makes up half of the overall hypotenuse, so we then multiply the result by    to obtain the final answer. In the following solution,    is the length of the base of one of the right triangles,    is the length of the two equivalent sides, and    is the length of the hypotenuse: