GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #581 : Problem Solving Questions

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

Possible Answers:

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

The triangle must be isosceles and obtuse.

The triangle must be scalene 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 $20 ^{\circ }$ $\displaystyle 20 ^{\circ }$, the third must have measure $180 - 20 - 20 = 140^{\circ } > 90^{\circ }$ $\displaystyle 180 - 20 - 20 = 140^{\circ } > 90^{\circ }$. 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 #13 : Calculating An Angle In An Acute / Obtuse Triangle

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

$\displaystyle x-y = 22$.

Evaluate $\displaystyle y$.

Possible Answers:

y = 43 $\displaystyle y = 43$

y = 65 $\displaystyle y = 65$

y = 50 $\displaystyle y = 50$

y = 25 $\displaystyle y = 25$

y = 47 $\displaystyle y = 47$

Correct answer:

y = 43 $\displaystyle y = 43$

Explanation:

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

$\displaystyle x+y + 72 = 180$

$\displaystyle x+y + 72 - 72 = 180 - 72$

$\displaystyle x+y = 108$

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

$\displaystyle x-y = 22$

$\displaystyle -(x-y )= - (22)$

$\displaystyle -x+y = -22$

$\underline{x+y = 108}$ $\displaystyle \underline{x+y = 108}$

2y = 86 $\displaystyle 2y = 86$

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

y = 43 $\displaystyle y = 43$

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

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

$\displaystyle x - y = 32$.

Which of the following is closest to $\displaystyle x$?

Possible Answers:

$\displaystyle 40$

$\displaystyle 50$

$\displaystyle 80$

$\displaystyle 60$

$\displaystyle 70$

Correct answer:

$\displaystyle 50$

Explanation:

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

$\displaystyle x+y+2x = 180$, or

$\displaystyle 3x+y = 180$

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

$\displaystyle 3x+y = 180$

$\underline{x - y = 32}$ $\displaystyle \underline{x - y = 32}$

$\displaystyle 4x$ =212 $\displaystyle =212$

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

x = 53 $\displaystyle x = 53$

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

Possible Answers:

$21^{\circ }$ $\displaystyle 21^{\circ }$

None of the other responses gives the correct answer.

$69^{\circ }$ $\displaystyle 69^{\circ }$

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

$111^{\circ }$ $\displaystyle 111^{\circ }$

Correct answer:

$111^{\circ }$ $\displaystyle 111^{\circ }$

Explanation:

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

$\displaystyle x+y+42 = 180$, or, equivalently,

$\displaystyle x+y+42 - 42 = 180 - 42$

$\displaystyle x+ y = 138$

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

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

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

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

By substitution,

$\displaystyle 69 + Z = 180$

and

$\displaystyle Z = 180 - 69 = 111$.

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

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

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

Possible Answers:

$146^{\circ }$ $\displaystyle 146^{\circ }$

$34^{\circ }$ $\displaystyle 34^{\circ }$

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

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

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

Correct answer:

$34^{\circ }$ $\displaystyle 34^{\circ }$

Explanation:

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

$\displaystyle x+y+107= 180$, or, equivalently,

$\displaystyle x+y+107 - 107= 180 - 107$

$\displaystyle x+ y = 73$

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

$\displaystyle 2x+2y = 146$

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

$\displaystyle 2x+2y +Z = 180$

By substitution,

$\displaystyle 146+Z = 180$

$\displaystyle Z = 180-146 = 34$

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

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

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

Express $\displaystyle z$ in terms of $\displaystyle y$.

Possible Answers:

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

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

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

$\displaystyle z = -3y + 180$

$\displaystyle z = - 2 y + 180$

Correct answer:

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

Explanation:

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

$\displaystyle x+x + y = 180$

$\displaystyle 2x+y = 180$

$\displaystyle 2x= 180 - y$

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

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

From Triangle 2, we can deduce that

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

$\displaystyle 3x+ z = 180$

$\displaystyle z = 180 -3x$

By substitution:

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

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

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

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Example Question #349 : Geometry

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

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

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

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 $180 ^{\circ }$ $\displaystyle 180 ^{\circ }$;

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

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

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

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

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

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

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Example Question #581 : Problem Solving Questions

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

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

Statement 1: $\angle 1$ $\displaystyle \angle 1$ is acute.

Statement 2: $\angle B$ $\displaystyle \angle B$ and $\angle C$ $\displaystyle \angle C$ 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 $\angle 1$ $\displaystyle \angle 1$ forms a linear pair with its interior angle $\angle B AC$ $\displaystyle \angle B AC$. Either both are right, or one is acute and one is obtuse. From Statement 1 alone, since $\angle 1$ $\displaystyle \angle 1$ is acute, $\angle B AC$ $\displaystyle \angle B AC$ is obtuse, and $\bigtriangleup ABC$ $\displaystyle \bigtriangleup ABC$ is an obtuse triangle.

Statement 2 alone is insufficient. Every triangle has at least two acute angles, and Statement 2 only establishes that $\angle B$ $\displaystyle \angle B$ and $\angle C$ $\displaystyle \angle C$ are both acute; the third angle, $\angle B AC$ $\displaystyle \angle B AC$, can be acute, right, or obtuse, so the question of whether $\bigtriangleup ABC$ $\displaystyle \bigtriangleup ABC$ 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

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

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

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

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

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

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

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Example Question #21 : Calculating An Angle In An Acute / Obtuse Triangle

The measures of the angles of one triangle, in degrees, are $\displaystyle X , 2X, Y$.

The measures of the angles of a second triangle, in degrees, are $\displaystyle Y + 10, Y + 20, 2X-10$.

What is $\displaystyle Y$?

Possible Answers:

Y = 30 $\displaystyle Y = 30$

Y = 60 $\displaystyle Y = 60$

Y = 40 $\displaystyle Y = 40$

Y = 20 $\displaystyle Y = 20$

Y = 50 $\displaystyle Y = 50$

Correct answer:

Y = 30 $\displaystyle Y = 30$

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:

$\displaystyle X+2X+ Y = 180$

$\displaystyle 3X+ Y = 180$

From the second:

$\displaystyle (Y + 10)+( Y + 20)+ (2X-10 )= 180$

$\displaystyle 2X+ 2Y + 20 = 180$

$\displaystyle 2X+ 2Y + 20 - 20 = 180 - 20$

$\displaystyle 2X+ 2Y = 160$

$(2X+ 2Y)\div 2 = 160 \div 2$ $\displaystyle (2X+ 2Y)\div 2 = 160 \div 2$

$\displaystyle X+Y = 80$

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

$\displaystyle X+Y = 80$

$\displaystyle -(X+Y) = - (80)$

$\displaystyle -X-Y = -80$

$\underline{3X+ Y = 180}$ $\displaystyle \underline{3X+ Y = 180}$

$\displaystyle 2X$ = 100 $\displaystyle = 100$

X = 50 $\displaystyle X = 50$

$\displaystyle X+Y = 80$, so

$\displaystyle 50 +Y = 80$

and Y = 30 $\displaystyle Y = 30$.

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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #581 : Problem Solving Questions

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

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

Example Question #346 : Geometry

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

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

Example Question #349 : Geometry

Example Question #581 : Problem Solving Questions

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

Example Question #21 : Calculating An Angle In An Acute / Obtuse Triangle

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