GMAT Math : Calculating an angle in an acute / obtuse triangle

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #11 : Acute / Obtuse Triangles

In the following triangle:

The angle  degrees

The angle  degrees

Angle1

(Figure not drawn on scale)

Find the value of .

Possible Answers:

Correct answer:

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)

Angle2

Example Question #91 : Triangles

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

Possible Answers:

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.

Correct answer:

The triangle must be isosceles and obtuse.

Explanation:

The sum of the measures of three angles of any triangle is 180; therefore, if two angles have measure , the third must have measure . This makes the triangle obtuse. Also, since the triangle has two congruent angles, it is isosceles by the Converse of the Isosceles Triangle Theorem.

Example Question #13 : Calculating An Angle In An Acute / Obtuse Triangle

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 #14 : Calculating An Angle In An Acute / Obtuse Triangle

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 #346 : Geometry

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 #15 : Calculating An Angle In An Acute / Obtuse Triangle

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 #16 : Calculating An Angle In An Acute / Obtuse Triangle

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 #349 : Geometry

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

Statement 1: 

Statement 2: 

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.

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.

BOTH STATEMENTS TOGETHER do 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 #581 : Problem Solving Questions

 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:

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.

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.

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 #11 : Calculating An Angle In An Acute / Obtuse Triangle

, 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 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:

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. 

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