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

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Example Question #342 : Geometry

In the following triangle:

AD = BD = CD

The angle DCB = 60 degrees

The angle DCA = 15 degrees

Angle1

(Figure not drawn on scale)

Find the value of .

Possible Answers:

7.5

Correct answer:

Explanation:

Since AD = BD = CD , the following triangles are isoscele: ADC, BDC, BDA .

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

The angle CBD = 60 degrees

The angle DAC = 15 degrees, and

The angle DBA = DAB = x degrees

Therefore:

The angle BAC = x + 15

The angle ACB = 60 + 15 = 75 degrees, and

The angle CBA = x + 60

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

BAC + ACB + CBA = 180 . So:

x + 15 + 75 + x + 60 = 180 .

Now let us solve the equation for x:

$2x + 150 = 180 \rightarrow x = 15$

(See image below - not drawn on scale)

Angle2

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Example Question #341 : Geometry

Which of the following is true of a triangle with two $20 ^{\circ }$ angles?

Possible Answers:

The triangle must be scalene and obtuse.

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

The triangle must be isosceles and obtuse.

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

The triangle must be isosceles and acute.

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 $20 ^{\circ }$ , the third must have measure $180 - 20 - 20 = 140^{\circ } > 90^{\circ }$ . This makes the triangle obtuse. Also, since the triangle has two congruent angles, it is isosceles by the Converse of the Isosceles Triangle Theorem.

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Example Question #344 : Geometry

The measures of the interior angles of a triangle are $x^{\circ }$ , $y^{\circ }$ , and $72^{\circ }$ . Also,

x-y = 22 .

Evaluate .

Possible Answers:

y = 65

y = 50

y = 47

y = 43

y = 25

Correct answer:

y = 43

Explanation:

The measures of the interior angles of a triangle have sum $180^{\circ }$ , so

x+y + 72 = 180

x+y + 72 - 72 = 180 - 72

x+y = 108

Along with x-y = 22 , a system of linear equations is formed that can be solved by adding:

x-y = 22

-(x-y )= - (22)

-x+y = -22

$\underline{x+y = 108}$

2y = 86

$2y \div 2 = 86 \div 2$

y = 43

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Example Question #345 : Geometry

The interior angles of a triangle have measures $x^{\circ }$ , $y^{\circ }$ , and $2x^{\circ }$ . Also,

x - y = 32 .

Which of the following is closest to ?

Possible Answers:

Correct answer:

Explanation:

The measures of the interior angles of a triangle have sum $180^{\circ }$ , so

x+y+2x = 180 , or

3x+y = 180

Along with x - y = 32 , a system of linear equations is formed that can be solved by adding:

3x+y = 180

$\underline{x - y = 32}$

=212

$4x \div 4 = 212 \div 4$

x = 53

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

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Example Question #346 : Geometry

A triangle has interior angles whose measures are $x^{\circ }$ , $y^{\circ }$ , and $42 ^{\circ }$ . A second triangle has interior angles, two of whose measures are $\frac{1}{2}x^{\circ }$ and $\frac{1}{2}y^{\circ }$ . What is the measure of the third interior angle of the second triangle?

Possible Answers:

$21^{\circ }$

None of the other responses gives the correct answer.

$69^{\circ }$

$84^{\circ }$

$111^{\circ }$

Correct answer:

$111^{\circ }$

Explanation:

The measures of the interior angles of a triangle have sum $180^{\circ }$ , so

x+y+42 = 180 , or, equivalently,

x+y+42 - 42 = 180 - 42

x+ y = 138

$\frac{1}{2}\left (x+ y \right )= \frac{1}{2} \cdot 138$

$\frac{1}{2} x+ \frac{1}{2} y =69$

$\frac{1}{2}x^{\circ }$ and $\frac{1}{2}y^{\circ }$ are the measures of two interior angles of the second triangle, so if we let be the measure of the third angle, then

$\frac{1}{2} x+ \frac{1}{2} y +Z= 180$

By substitution,

69 + Z = 180

and

Z = 180 - 69 = 111 .

The correct response is $111^{\circ }$ .

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Example Question #91 : Triangles

The measures of the interior angles of Triangle 1 are $x^{\circ }$ , $y^{\circ }$ , and $107^{\circ }$ . The measures of two of the interior angles of Triangle 2 are $2x^{\circ }$ and $2y^{\circ }$ . Which of the following is the measure of the third interior angle of Triangle 2?

Possible Answers:

$146^{\circ }$

$143\frac{1}{2 }^{\circ }$

$36\frac{1}{2} ^{\circ }$

$53\frac{1}{2}^{\circ }$

$34^{\circ }$

Correct answer:

$34^{\circ }$

Explanation:

The measures of the interior angles of a triangle have sum $180^{\circ }$ , so

x+y+107= 180 , or, equivalently,

x+y+107 - 107= 180 - 107

x+ y = 73

$2 (x+ y )= 2 \cdot 73$

2x+2y = 146

$2x^{\circ }$ and $2y^{\circ }$ are the measures of two interior angles of the second triangle, so if we let be the measure of the third angle, then

2x+2y +Z = 180

By substitution,

146+Z = 180

Z = 180-146 = 34

The correct response is $34^{\circ }$ .

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Example Question #348 : Geometry

Triangle 1 has three interior angles with measures $x^{\circ }$ , $x^{\circ }$ , and $y^{\circ }$ . Triangle 1 has three interior angles with measures $\frac{3}{2} x^{\circ }$ , $\frac{3}{2} x^{\circ }$ , and $z^{\circ }$ .

Express in terms of .

Possible Answers:

$z = \frac{3}{2} y - 90$

z = -3y + 180

z = - 2 y + 180

$z = \frac{2}{3} y$

$z = \frac{1}{6} y +150$

Correct answer:

$z = \frac{3}{2} y - 90$

Explanation:

The sum of the measures of the interior angles of a triangle is $180^{\circ }$ , so it can be determined from Triangle 1 that

x+x + y = 180

2x+y = 180

2x= 180 - y

$\frac{1}{2} \cdot 2x = \frac{1}{2} \cdot (180-y)$

$x =90-\frac{1}{2} y$

From Triangle 2, we can deduce that

$\frac{3}{2} x^{\circ } + \frac{3}{2} x^{\circ } + z = 180$

3x+ z = 180

z = 180 -3x

By substitution:

$z = 180 -3\left ( 90-\frac{1}{2} y \right )$

$z = 180 -270 +\frac{3}{2} y$

$z = \frac{3}{2} y - 90$

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Example Question #349 : Geometry

Is $\bigtriangleup ABC$ an acute triangle, a right triangle, or an obtuse triangle?

Statement 1: $m \angle A + m \angle B < 90 ^{\circ }$

Statement 2: $(AC)^{2} + (BC ) ^{2}< (AB) ^{2}$

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 $180 ^{\circ }$ ;

$m \angle A + m \angle B < 90 ^{\circ }$ , or, equivalently, for some positive number ,

$m \angle A + m \angle B =( 90 - N) ^{\circ }$ ,

$m \angle A + m \angle B + m \angle C = 180 ^{\circ }$

$( 90 - N) ^{\circ }+ m \angle C = 180 ^{\circ }$

$m \angle C = 180 ^{\circ } - ( 90 - N) ^{\circ } = ( 90+N) ^{\circ }$

Therefore, $m \angle C > 90 ^{\circ }$ , making $\angle C$ obtuse, and $\bigtriangleup ABC$ 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 $\bigtriangleup ABC$ is an obtuse triangle.

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

$\angle 1$ is an exterior angle of $\bigtriangleup ABC$ at $\angle B AC$ .

Is $\bigtriangleup ABC$ an acute triangle, a right triangle, or an obtuse triangle?

Statement 1: $\angle 1$ is acute.

Statement 2: $\angle B$ and $\angle C$ 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.

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.

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.

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 $\angle 1$ forms a linear pair with its interior angle $\angle B AC$ . Either both are right, or one is acute and one is obtuse. From Statement 1 alone, since $\angle 1$ is acute, $\angle B AC$ is obtuse, and $\bigtriangleup ABC$ is an obtuse triangle.

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

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

$\angle 1$ , $\angle 2$ , and $\angle 3$ are all exterior angles of $\bigtriangleup ABC$ with vertices , , and , respectively.

Is $\bigtriangleup ABC$ an acute triangle, a right triangle, or an obtuse triangle?

Statement 1: $\angle 1$ , $\angle 2$ , and $\angle 3$ are all obtuse angles.

Statement 2: $\angle A \cong \angle B$ .

Note: For purposes of this problem, $\angle A$ , $\angle B$ , and $\angle C$ 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 $180^{\circ }$ , must be two right angles or one acute angle and one obtuse angle. Since $\angle 1$ , $\angle 2$ , and $\angle 3$ are all obtuse angles, it follows that their respective interior angles - the three angles of $\bigtriangleup ABC$ - are all acute. This makes $\bigtriangleup ABC$ an acute triangle.

Statement 2 alone provides insufficient information to answer the question. For example, if $\angle A$ and $\angle B$ each measure $10^{\circ }$ and $\angle C$ measures $160^{\circ }$ , the sum of the angle measures is $180^{\circ }$ , $\angle A$ and $\angle B$ are congruent, and $\angle C$ is an obtuse angle (measuring more than $90^{\circ }$ ); this makes $\bigtriangleup ABC$ an obtuse triangle. But if $\angle A$ , $\angle B$ , and $\angle C$ each measure $60^{\circ }$ , the sum of the angle measures is again $180^{\circ }$ , $\angle A$ and $\angle B$ are again congruent, and all three angles are acute (measuring less than $90^{\circ }$ ); this makes $\bigtriangleup ABC$ an acute triangle.

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

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #342 : Geometry

Example Question #341 : Geometry

Example Question #344 : Geometry

Example Question #345 : Geometry

Example Question #346 : Geometry

Example Question #91 : Triangles

Example Question #348 : Geometry

Example Question #349 : Geometry

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

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

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