GMAT Math : Triangles

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Example Questions

Example Question #21 : Triangles

In $\bigtriangleup ABC$ , if $AB=k$ , $BC=k+3$ and $AC=m$ , which of the three angles of $\triangle ABC$ has the greatest degree measure?

(1) $k=3$

(2) $m=k+4$

Possible Answers:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Correct answer:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Explanation:

The longest side is opposite the largest angle for all triangles.

(1) Substituting 3 for $k$ means that $AB=3$ and $BC=5$ . But the value of $m$ given for side $AC$ is still unknown $\rightarrow$ NOT sufficient.

(2) Since $k+3>k$ , the longest side must be either $k+3$ or $m$ . So, knowing whether $m>k+3$ is sufficient.

If $m=k+4$ , knowing that $k+4>k+3$ ,

then $m>k+3$ $\rightarrow$ SUFFICIENT.

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Example Question #22 : Triangles

A triangle contains a $110^{\circ}$ angle. What are the other angles in the triangle?

(1) The triangle is isosceles.

(2)The triangle has a perimeter of 12.

Possible Answers:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

Correct answer:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Explanation:

Statement 1: An isosceles triangle has two equal angles. Since the interior angles of a triangle always sum to $180^{\circ}}$ , the only possible angles the other sides could have are $35^{\circ}$ .

Statement 2: This does not provide any information relevant to the question.

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

Triangle_b

Note: figure NOT drawn to scale.

$m \angle A = 46 ^{\circ }$ . What is $m \angle 1$ ?

Statement 1: $\overline{AB} \cong \overline{AC}$

Statement 2: $m \angle B = 67 ^{\circ }$

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.

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.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If $\overline{AB} \cong \overline{AC}$ , then by the Isosceles Triangle Theorem, $m \angle B = m \angle 2$ . Since the sum of the measures of a triangle is 180,

$m \angle A + m \angle B + m \angle 2 = 180$

After some substitution,

$46 + m \angle 2 + m \angle 2 = 180$

$46 + 2m \angle 2 = 180$

$2m \angle 2 = 134$

$m \angle 2 = 67$

Since $\angle 1$ and $\angle 2$ form a linear pair,

$m \angle 1 + m \angle 2 = 180$ , and

$m \angle 1 = 180 - m \angle 2 = 180 - 67 = 113^{\circ }$

If $m \angle B = 67 ^{\circ }$ , then by the Triangle Exterior Angle Theorem,

$m \angle 2 = m \angle A + m \angle B = 46 + 67 = 113^{\circ }$

So either statement by itself provides sufficient information.

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Example Question #4 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

In $\triangle PQR$ above, what is the value of ?

(1) x=1.5y

(2) x+y=112.5

Possible Answers:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

Correct answer:

EACH statement ALONE is sufficient.

Explanation:

There is an implied condition: 2x+y=180 . Therefore, with each statement, we have 2 unknown numbers and 2 equations. In this case, we can take a guess that we will be able to find the value of by using each statement alone. It’s better to check by actually solving this problem.

For statement (1), we can plug 1.5y into 2x+y=180 . Now we have 4y=180 , which means y=45 .

For statement (2), we can rewrite the equation to be x=112.5-y and then plug into 2x+y=180 , making it

$2\cdot \left ( 112.5-y \right )+y=180$

Then we can solve for and get y=45 .

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Example Question #5 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

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

Statement 1: $m\angle A + m\angle B = 120^{\circ }$

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

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

Each statement alone allows us to calculate the measure of one of the angles by subtracting the sum of the other two from 180.

From Statement 1:

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

$120^{\circ } + m\angle C = 180^{\circ }$

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

From Statement 2:

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

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

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

Neither statement alone is enough to answer the question, since either statement leaves enough angle measurement to allow one of the other triangles to be right or obtuse. But the two statements together allow us to calculate $m\angle B$ :

$70 ^{\circ }+ m\angle B + 60^{\circ }= 180^{\circ }$ :

$m\angle B = 50 ^{\circ}$

This allows us to prove $\Delta ABC$ acute.

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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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GMAT Math : Triangles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #21 : Triangles

Example Question #22 : Triangles

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

Example Question #4 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Example Question #5 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

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

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