GMAT Math : Geometry

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Example Questions

Example Question #6 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Is triangle $\Delta ABC$ acute, right, or obtuse?

Statement 1: $\angle A + \angle B = 80^{\circ }$

Statement 2: $\angle B+ \angle C = 110^{\circ }$

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

From Statement 2:

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

$80 ^{\circ }+ m \angle C = 180^{\circ }$

$m \angle C = 100^{\circ } > 90^{\circ}$

This is enough to prove the triangle is obtuse.

From Statement 2 we can calculate $m \angle A$ :

$m \angle A + 110 ^{\circ }= 180^{\circ }$

$m \angle A =70 ^{\circ }$

We present two cases to demonstrate that this is not enough information to answer the question:

$m \angle A =70 ^{\circ }, m \angle B = 90^{\circ } , m \angle C = 20 ^{\circ}$ - right triangle.

$m \angle A =70 ^{\circ },m \angle B = 80^{\circ } , m \angle C = 30 ^{\circ}$ - acute triangle.

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Example Question #383 : Data Sufficiency Questions

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

Statement 1: $\angle A$ is complementary to $\angle C$ .

Statement 2: The triangle has exactly two acute angles.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

If we assume Statement 1 alone, that $\angle A$ is complementary to $\angle C$ , then by definition, $m \angle A + m \angle C = 90 ^{\circ }$ . Since $m \angle A+ m \angle B + m \angle C = 180 ^{\circ }$ ,

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

$m \angle B +90 ^{\circ } = 180 ^{\circ }$

$m \angle B +90 ^{\circ } -90 ^{\circ } = 180 ^{\circ }-90 ^{\circ }$

$m \angle B =90 ^{\circ }$

This makes $\angle B$ a right angle and $\Delta ABC$ a right triangle.

Statement 2 alone is inufficient, however, since a triangle with exactly two acute angles can be either right or obtuse.

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Example Question #8 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. Is $\Delta ABC$ an equilateral triangle?

Statement 1: $m \widehat{ABC} = 240 ^{\circ }$

Statement 2: $m \widehat{ACB} = 260 ^{\circ }$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Each of the arcs mentioned in the statements is a major arc corresponding to one of these minor arcs, so, specifically, $m \widehat{A C}= 360 ^{\circ } - m \widehat{ABC}$ and $m \widehat{AB} = 360 ^{\circ } - m \widehat{ACB}$ .

From Statement 1 alone, we can calculate:

$m \widehat{A C}= 360 ^{\circ } - m \widehat{ABC} = 360 ^{\circ }- 240 ^{\circ } = 120 ^{\circ }$

This does not prove or disprove $\Delta ABC$ to be equilateral, since one minor arc can measure $120 ^{\circ }$ without the other two doing so.

From Statement 2 alone, we can calculate

$m \widehat{AB} = 360 ^{\circ } - 260^{\circ } = 100 ^{\circ }$

so we know that $\Delta ABC$ is not equilateral.

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Example Question #2501 : Gmat Quantitative Reasoning

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. Is $\Delta ABC$ an equilateral triangle?

Statement 1: $m \widehat{AB} = 120 ^{\circ }$

Statement 2: $m \widehat{BC} = 140 ^{\circ }$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

The measure of each of the three angles of the triangle, being angles inscribed in the circle, is one-half the measure of the arc it intercepts. For the triangle to be equilateral, each angle has to measure $60^{\circ }$ , and $m \widehat{AB} = m \widehat{BC} =m \widehat{AC} =120 ^{\circ }$ . This is neither proved nor diproved by Statement 1 alone, since one arc can measure $120 ^{\circ }$ without the other two doing so; it is, however, disproved by Statement 2 alone.

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

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

Statement 1: There are exactly two acute angles.

Statement 2: The exterior angles of the triangle at vertex are both acute.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 tells us that the triangle is either right or obtuse, but nothing more.

Statement 2 tells us that the triangle is obtuse. An exterior angle of a triangle is supplemetary to the interior angle to which it is adjacent. Since the supplement of an acute angle is obtuse, this means the triangle must have an obtuse angle.

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

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

Statement 1: $\angle A$ and $\angle B$ are both acute.

Statement 2: $\angle A$ and $\angle C$ are both acute.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Every triangle has at least two acute angles, so neither statement is sufficient to answer the question. The two statements together, however, are enough to prove $\Delta ABC$ to have three acute angles and to therefore be an acute triangle.

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Example Question #31 : Triangles

Is $\Delta ABC$ an isosceles triangle?

Statement 1: $m \angle A + m \angle B = 100 ^{\circ }$

Statement 2: $m \angle A + m \angle C = 100 ^{\circ }$

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

From Statement 1 it can be deduced that $m \angle C = 180^{\circ } -\left ( m \angle A + m \angle B \right ) = 180^{\circ } - 100^{\circ } = 80^{\circ }$ . Similarly, from Statement 2 it can be deduced that $m \angle B = 80^{\circ }$ . Neither statement alone gives information about the other two angles. Both statements together, however, prove that $\angle B \cong \angle C$ , making the triangle isosceles by the Isosceles Triangle Theorem.

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Example Question #271 : Geometry

True or false: $\Delta ABC$ is equilateral.

Statement 1: $\angle A \ncong \angle B$

Statement 2: $m \angle A =90^{\circ }$

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

An equilateral triangle has three congruent angles, each of which measure $60^{\circ }$ . Both statements contradict this condition, proving that $\Delta ABC$ is not equilateral.

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Example Question #32 : Triangles

True or false: $\Delta ABC$ is equilateral.

Statement 1: $\angle A \cong \angle B$

Statement 2: $m \angle C =60^{\circ }$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

An equilateral triangle has three congruent angles, each of which measure $60^{\circ }$ . Statement 1 alone establishes the congruence of two angles but not the third; for example, the triangle could be 45-45-90 and fit the condition. Statement 2 alone only establishes the measure of one angle.

Assume both statements are true. The degree measures of the angles of a triangle add up to $180^{\circ}$ , and, since $\angle A \cong \angle B$ , we can set up and solve:

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

$m \angle A + m \angle A+ 60^{\circ} = 180^{\circ}$

$2m \angle A= 120^{\circ}$

$m \angle A= 60^{\circ}$

$\angle A \cong \angle B$ , so $m \angle A = m \angle B = m \angle C =60^{\circ }$ in $\Delta ABC$ .

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Example Question #272 : Geometry

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

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

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

Statement 2: $m \angle B + m \angle C = 80 ^{\circ }$

Possible Answers:

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.

EITHER STATEMENT ALONE provides 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.

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:

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 also provides sufficient information; the sum of the measures of interior angles of a triangle is $180 ^{\circ }$ ; since the sum of the measures of two of them, $m \angle B$ and $m \angle C$ , is $80 ^{\circ }$ , the other angle, $\angle B AC$ , has measure $180 ^{\circ } - 80 ^{\circ } = 100 ^{\circ } > 90 ^{\circ }$ , making $\angle B AC$ obtuse, and making $\bigtriangleup ABC$ an obtuse triangle.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #6 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Example Question #383 : Data Sufficiency Questions

Example Question #8 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Example Question #2501 : Gmat Quantitative Reasoning

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

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

Example Question #31 : Triangles

Example Question #271 : Geometry

Example Question #32 : Triangles

Example Question #272 : Geometry

All GMAT Math Resources

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