GMAT Math : Acute / Obtuse Triangles

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Example Question #1 : Calculating An Angle In An Acute / Obtuse Triangle

Two angles of an isosceles triangle measure $(x + 20) ^{\circ}$ and $\left ( 100-x\right )^{ \circ}$ . What are the possible values of ?

Possible Answers:

$x = 40 \textrm{ or }x = 80$

x = 40

x = 60

x = 80

$x = 60 \textrm{ or } x = 80$

Correct answer:

x = 40

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.

x + 20 = 100 - x

x+ 20 + x = 100 - x + x

2x+ 20 = 100

2x + 20 - 20= 100 -20

2x = 80

$2x \div 2 =80 \div 2$

x = 40

The angle measures are $60 ^{\circ}, 60 ^{\circ}, 60 ^{\circ}$ , 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 $(x + 20) ^{\circ}$ .

Then, since the sum of the angle measures is 180,

$\left (x + 20 \right ) +\left (x +20 \right )+ \left ( 100-x \right ) = 180$

x + 140= 180

x + 140 - 140 = 180- 140

x = 40

as before

Case 3: The third angle has measure $\left ( 100-x\right )^{ \circ}$

$\left (x + 20 \right ) +\left (100-x \right )+ \left ( 100-x \right ) = 180$

220 -x = 180

220 -x -220 = 180-220

-x = -40

x = 40

as before.

Thus, the only possible value of is 40.

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Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

Two angles of an isosceles triangle measure $\left (x + 30 \right )^{\circ }$ and $\left (x + 36 \right )^{\circ }$ . What are the possible value(s) of ?

Possible Answers:

x= 12

x=6

$x=26 \textrm{ or }x=28$

$x= 12, \; x=26 \textrm{, or }x=28$

$x= 6, \; x=26 \textrm{, or }x=28$

Correct answer:

$x=26 \textrm{ or }x=28$

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.

x + 30 = x + 36

x + 30-x = x + 36-x

30=36

This is a false statement, indicating that this situation is impossible.

Case 2: The third angle has measure $\left (x + 30 \right )^{\circ }$ .

Then, since the sum of the angle measures is 180,

$\left (x + 30 \right ) +\left (x + 30 \right )+ \left ( x + 36 \right ) = 180$

3x + 96 = 180

3x =84

x=28

This makes the angle measures $58^{\circ },58^{\circ },64^{\circ }$ , a plausible scenario.

Case 3: the third angle has measure $\left (x + 36 \right )^{\circ }$

Then, since the sum of the angle measures is 180,

$\left (x + 30 \right ) +\left (x + 36 \right )+ \left ( x + 36 \right ) = 180$

3x +102 = 180

3x =78

x=26

This makes the angle measures $56^{\circ }, 62^{\circ }, 62^{\circ }$ , a plausible scenario.

Therefore, either x=26 or x=28

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Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

$\Delta ABC \sim \Delta XYZ$

$m \angle A = 20 ^{\circ }; m \angle Y = 120 ^{\circ }$

Which of the following is true of $\Delta ABC$ ?

Possible Answers:

$\Delta ABC$ may be scalene or isosceles, but it is acute,

$\Delta ABC$ is isosceles and obtuse.

$\Delta ABC$ is scalene and acute.

$\Delta ABC$ may be scalene or isosceles, but it is obtuse.

$\Delta ABC$ is scalene and obtuse.

Correct answer:

$\Delta ABC$ is scalene and obtuse.

Explanation:

By similarity, $m \angle B = m \angle Y = 120 ^{\circ }$ .

Since measures of the interior angles of a triangle total $180 ^{\circ }$ ,

$m \angle A + m \angle B + m \angle C = 180$

$20 + 120 + m \angle C = 180$

$140 + m \angle C = 180$

$140 + m \angle C -140 = 180-140$

$m \angle C =40^{\circ }$

Since the three angle measures of $\Delta ABC$ are all different, no two sides measure the same; the triangle is scalene. Also, since $m \angle B = 120 ^{\circ } > 90^{\circ }$ , the angle is obtuse, and $\Delta ABC$ is an obtuse triangle.

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Example Question #1 : Calculating An Angle In An Acute / Obtuse Triangle

Which of the following is true of a triangle with three angles whose measures have an arithmetic mean of $90^{\circ }$ ?

Possible Answers:

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

The triangle cannot exist.

The triangle must be right and isosceles.

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

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

Correct answer:

The triangle cannot exist.

Explanation:

The sum of the measures of three angles of any triangle is 180; therefore, their mean is $180 \div 3 = 60$ , making a triangle with angles whose measures have mean 90 impossible.

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Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

Two angles of a triangle measure $47 ^{\circ }$ and $23^{\circ }$ . What is the measure of the third angle?

Possible Answers:

$70^{\circ }$

$20^{\circ }$

$160^{\circ }$

$35^{ \circ }$

$110^{\circ }$

Correct answer:

$110^{\circ }$

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:

$(180-47-23)^{\circ } = 110^{\circ }$

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Example Question #332 : Geometry

The angles of a triangle measure $x^{\circ }, x^{\circ },\left ( x -54 \right ) ^{\circ }$ . Evaluate .

Possible Answers:

x = 96

x = 112

x = 78

x = 84

x = 72

Correct answer:

x = 78

Explanation:

The sum of the measures of the angles of a triangle total $180^{\circ }$ , so we can set up and solve for in the following equation:

x + x + (x-54) = 180

3x-54 = 180

3x-54 + 54 = 180+ 54

3x =234

$3x\div 3 =234 \div 3$

x = 78

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Example Question #7 : Calculating An Angle In An Acute / Obtuse Triangle

An exterior angle of $\Delta ABC$ with vertex measures $150^{\circ }$ ; an exterior angle of $\Delta ABC$ with vertex measures $110^{\circ }$ . Which is the following is true of $\Delta ABC$ ?

Possible Answers:

$\Delta ABC$ is obtuse and isosceles

$\Delta ABC$ is acute and scalene

$\Delta ABC$ is right and scalene

$\Delta ABC$ is obtuse and scalene

$\Delta ABC$ is acute and isosceles

Correct answer:

$\Delta ABC$ is acute and scalene

Explanation:

An interior angle of a triangle measures $180 ^{\circ }$ minus the degree measure of its exterior angle. Therefore:

$m \angle A = 180 ^{\circ } - 150 ^{\circ } = 30 ^{\circ }$

$m \angle B= 180 ^{\circ } - 110 ^{\circ } = 70 ^{\circ }$

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

$m \angle C = 180 ^{\circ } - (m \angle A + m \angle B) = 180 ^{\circ } - (30 ^{\circ }+ 70 ^{\circ }) = 180 ^{\circ } -100 ^{\circ } = 80 ^{\circ }$ .

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.

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Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram.

$m\angle DCF = 140 ^{\circ } , m \angle ABF = 134 ^{\circ }$

Evaluate $m \angle EFB$ .

Possible Answers:

$84^{\circ }$

$96^{\circ }$

$86^{\circ }$

$74^{\circ }$

$76^{\circ }$

Correct answer:

$86^{\circ }$

Explanation:

The sum of the exterior angles of a triangle, one per vertex, is $360 ^{\circ}$ . $\angle DCF$ , $\angle ABF$ and $\angle EFB$ are exterior angles at different vertices, so

$m \angle EFB + m\angle DCF + m \angle ABF = 360 ^{\circ }$

$m \angle EFB + 140 ^{\circ } + 134 ^{\circ } = 360 ^{\circ }$

$m \angle EFB + 274 ^{\circ } = 360 ^{\circ }$

$m \angle EFB + 274 ^{\circ }- 274 ^{\circ } = 360 ^{\circ } - 274 ^{\circ }$

$m \angle EFB = 86^{\circ }$

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Example Question #11 : Acute / Obtuse Triangles

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 #12 : Acute / Obtuse Triangles

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

Possible Answers:

The triangle must be isosceles and obtuse.

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

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 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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GMAT Math : Acute / Obtuse Triangles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Calculating An Angle In An Acute / Obtuse Triangle

Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

Example Question #1 : Calculating An Angle In An Acute / Obtuse Triangle

Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

Example Question #332 : Geometry

Example Question #7 : Calculating An Angle In An Acute / Obtuse Triangle

Example Question #2 : Calculating An Angle In An Acute / Obtuse Triangle

Example Question #11 : Acute / Obtuse Triangles

Example Question #12 : Acute / Obtuse Triangles

All GMAT Math Resources

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