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

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Example Questions

Example Question #3 : Acute / Obtuse Triangles

Which of the following cannot be the measure of a base angle of an isosceles triangle?

Possible Answers:

$50^{\circ }$ $\displaystyle 50^{\circ }$

$100 ^{\circ }$ $\displaystyle 100 ^{\circ }$

$1 ^{\circ }$ $\displaystyle 1 ^{\circ }$

Each of the other choices can be the measure of a base angle of an isosceles triangle.

$80 ^{\circ }$ $\displaystyle 80 ^{\circ }$

Correct answer:

$100 ^{\circ }$ $\displaystyle 100 ^{\circ }$

Explanation:

An isosceles triangle has two congruent angles by the Isosceles Triangle Theorem; these are the base angles. Since at least two angles of any triangle must be acute, a base angle must be acute - that is, it must measure under $90^{\circ }$ $\displaystyle 90^{\circ }$. The only choice that does not fit this criterion is $100 ^{\circ }$ $\displaystyle 100 ^{\circ }$, making this the correct choice.

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Example Question #4 : Acute / Obtuse Triangles

Let the three interior angles of a triangle measure x, x+30 $\displaystyle x, x+30$, and 2x - 10 $\displaystyle 2x - 10$. Which of the following statements is true about the triangle?

Possible Answers:

The triangle is isosceles and obtuse.

The triangle is isosceles and acute.

The triangle is scalene and acute.

The triangle is scalene and obtuse.

The triangle is scalene and right.

Correct answer:

The triangle is isosceles and acute.

Explanation:

If these are the measures of the interior angles of a triangle, then they total $180^{\circ }$ $\displaystyle 180^{\circ }$. Add the expressions, and solve for $\displaystyle x$.

$x +\left ( x + 30 \right ) +\left ( 2x - 10 \right ) = 180$ $\displaystyle x +\left ( x + 30 \right ) +\left ( 2x - 10 \right ) = 180$

$\displaystyle 4x +20 = 180$

$\displaystyle 4x = 160$

x = 40 $\displaystyle x = 40$

One angle measures $40^{\circ }$ $\displaystyle 40^{\circ }$. The others measure:

$x + 30 = 40 + 30 = 70^{\circ }$ $\displaystyle x + 30 = 40 + 30 = 70^{\circ }$

$2x-10 = 2 \cdot 40 - 10 = 80 - 10 = 70 ^{\circ }$ $\displaystyle 2x-10 = 2 \cdot 40 - 10 = 80 - 10 = 70 ^{\circ }$

All three angles measure less than $90 ^{\circ }$ $\displaystyle 90 ^{\circ }$ , so the triangle is acute. Also, there are two congruent angles, so by the converse of the Isosceles Triangle Theorem, two sides are congruent, and the triangle is isosceles.

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Example Question #571 : Gmat Quantitative Reasoning

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

Possible Answers:

x = 40 $\displaystyle x = 40$

x = 60 $\displaystyle x = 60$

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

x = 80 $\displaystyle x = 80$

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

Correct answer:

x = 40 $\displaystyle x = 40$

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.

$\displaystyle x + 20 = 100 - x$

$\displaystyle x+ 20 + x = 100 - x + x$

$\displaystyle 2x+ 20 = 100$

$\displaystyle 2x + 20 - 20= 100 -20$

2x = 80 $\displaystyle 2x = 80$

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

x = 40 $\displaystyle x = 40$

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

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

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

$\displaystyle x + 140= 180$

$\displaystyle x + 140 - 140 = 180- 140$

x = 40 $\displaystyle x = 40$

as before

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

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

$\displaystyle 220 -x = 180$

$\displaystyle 220 -x -220 = 180-220$

$\displaystyle -x = -40$

x = 40 $\displaystyle x = 40$

as before.

Thus, the only possible value of $\displaystyle x$ is 40.

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Example Question #3 : Calculating An Angle In An Acute / Obtuse Triangle

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

Possible Answers:

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

x= 12 $\displaystyle x= 12$

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

x=6 $\displaystyle x=6$

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

Correct answer:

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

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.

$\displaystyle x + 30 = x + 36$

$\displaystyle x + 30-x = x + 36-x$

30=36 $\displaystyle 30=36$

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

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

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

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

$\displaystyle 3x + 96 = 180$

3x =84 $\displaystyle 3x =84$

x=28 $\displaystyle x=28$

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

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

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

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

$\displaystyle 3x +102 = 180$

3x =78 $\displaystyle 3x =78$

x=26 $\displaystyle x=26$

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

Therefore, either x=26 $\displaystyle x=26$ or x=28 $\displaystyle x=28$

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

$\Delta ABC \sim \Delta XYZ$ $\displaystyle \Delta ABC \sim \Delta XYZ$

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

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

Possible Answers:

$\Delta ABC$ $\displaystyle \Delta ABC$ is isosceles and obtuse.

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

$\Delta ABC$ $\displaystyle \Delta ABC$ is scalene and acute.

$\Delta ABC$ $\displaystyle \Delta ABC$ is scalene and obtuse.

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

Correct answer:

$\Delta ABC$ $\displaystyle \Delta ABC$ is scalene and obtuse.

Explanation:

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

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

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

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

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

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

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

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

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Example Question #11 : Acute / Obtuse Triangles

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

Possible Answers:

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

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

The triangle cannot exist.

The triangle must be right and isosceles.

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

Correct answer:

The triangle cannot exist.

Explanation:

The sum of the measures of three angles of any triangle is 180; therefore, their mean is $180 \div 3 = 60$ $\displaystyle 180 \div 3 = 60$, 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 $47 ^{\circ }$ $\displaystyle 47 ^{\circ }$ and $23^{\circ }$ $\displaystyle 23^{\circ }$. What is the measure of the third angle?

Possible Answers:

$20^{\circ }$ $\displaystyle 20^{\circ }$

$35^{ \circ }$ $\displaystyle 35^{ \circ }$

$160^{\circ }$ $\displaystyle 160^{\circ }$

$110^{\circ }$ $\displaystyle 110^{\circ }$

$70^{\circ }$ $\displaystyle 70^{\circ }$

Correct answer:

$110^{\circ }$ $\displaystyle 110^{\circ }$

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:

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

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

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

Possible Answers:

x = 78 $\displaystyle x = 78$

x = 112 $\displaystyle x = 112$

x = 84 $\displaystyle x = 84$

x = 96 $\displaystyle x = 96$

x = 72 $\displaystyle x = 72$

Correct answer:

x = 78 $\displaystyle x = 78$

Explanation:

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

$\displaystyle x + x + (x-54) = 180$

$\displaystyle 3x-54 = 180$

$\displaystyle 3x-54 + 54 = 180+ 54$

3x =234 $\displaystyle 3x =234$

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

x = 78 $\displaystyle x = 78$

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

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

Possible Answers:

$\Delta ABC$ $\displaystyle \Delta ABC$ is obtuse and isosceles

$\Delta ABC$ $\displaystyle \Delta ABC$ is acute and scalene

$\Delta ABC$ $\displaystyle \Delta ABC$ is right and scalene

$\Delta ABC$ $\displaystyle \Delta ABC$ is obtuse and scalene

$\Delta ABC$ $\displaystyle \Delta ABC$ is acute and isosceles

Correct answer:

$\Delta ABC$ $\displaystyle \Delta ABC$ is acute and scalene

Explanation:

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

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

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

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

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

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 #11 : Acute / Obtuse Triangles

Lines

Note: Figure NOT drawn to scale.

Refer to the above diagram.

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

Evaluate $m \angle EFB$ $\displaystyle m \angle EFB$.

Possible Answers:

$86^{\circ }$ $\displaystyle 86^{\circ }$

$74^{\circ }$ $\displaystyle 74^{\circ }$

$76^{\circ }$ $\displaystyle 76^{\circ }$

$84^{\circ }$ $\displaystyle 84^{\circ }$

$96^{\circ }$ $\displaystyle 96^{\circ }$

Correct answer:

$86^{\circ }$ $\displaystyle 86^{\circ }$

Explanation:

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

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

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

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

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

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

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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 #3 : Acute / Obtuse Triangles

Example Question #4 : Acute / Obtuse Triangles

Example Question #571 : Gmat Quantitative Reasoning

Example Question #3 : Calculating An Angle In An Acute / Obtuse Triangle

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

Example Question #11 : Acute / Obtuse Triangles

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

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

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

Example Question #11 : Acute / Obtuse Triangles

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