GMAT Math : Acute / Obtuse Triangles

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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 #12 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

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 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:

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 #13 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

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

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

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

Possible Answers:

EITHER statement ALONE is 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 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.

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 #14 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

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

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

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

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.

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:

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 #15 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

$\angle 1$ 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:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do 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.

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 provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides 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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Example Question #16 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

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

Statement 1: AB + BC > AC

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

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.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do 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.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Statement 1 is true for any triangle $\bigtriangleup ABC$ by the Triangle Inequality, which states that the sum of the lengths of any two sides is greater than that of the third. Therefore, Statement 1 provides unhelpful information.

Statement 2 alone, however, proves that $\bigtriangleup ABC$ is obtuse, since the sum of the squares of the lengths of two sides exceeds the square of the length of the third.

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Example Question #17 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

A triangle has an interior angle of measure $30^{\circ }$ . Give the measures of the other two angles.

Statement 1: The triangle is isosceles.

Statement 2: The triangle is obtuse.

Possible Answers:

BOTH statements TOGETHER are insufficient 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.

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.

Correct answer:

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

Explanation:

Knowing only the triangle is obtuse only tells you that there is one obtuse angle, but along with the fact that there is a $30^{\circ }$ angle, this allows no further conclusions.

Knowing only that the triangle is isosceles, you can deduce from the Isosceles Triangle Theorem that there are two angles of equal measure; as the measures of the three angles are $180^{\circ }$ , there are two possibilities: the triangle is a $30^{\circ }-30^{\circ }-120^{\circ }$ triangle, or it is a $30^{\circ }-75^{\circ }-75^{\circ }$ triangle, but you cannot choose between the two without further information.

Knowing both facts allows you to choose the first of those two options.

The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

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Example Question #1 : Dsq: Calculating The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

Find the hypotenuse of an obtuse triangle.

Statement 1: Two given lengths with an inscribed angle.

Statement 2: Two known angles.

Possible Answers:

$\textup{EACH statement ALONE is sufficient.}$

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

$\textup{BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question. }$

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

Correct answer:

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

Explanation:

Statement 1: Two given lengths with an inscribed angle.

Draw a picture of the scenario. The values of $\textup{a}$ , $\textup{b}$ , and angle $\textup{C}$ are known values.

Use the Law of Cosines to determine side length $\textup{c}$ .

c^2=a^2+b^2-2abcos(C)

Statement 2: Two known angles.

There is insufficient information to solve for the length of the hypotenuse with only two interior angles. The third angle can be determined by subtracting the 2 angles from 180 degrees.

The triangle can be enlarged or shrunk to any degree with any scale factor and still yield the same interior angles. There must also be at least 1 side length in order to calculate the hypotenuse of the triangle by the Law of Cosines.

Therefore:

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

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Example Question #2 : Dsq: Calculating The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

Find the length of the hypotenuse of obtuse triangle TLC:

I) $\angle T=30^{\circ}=2* \angle C$

II) Side T is 3cm

Possible Answers:

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

Find the length of the hypotenuse of obtuse triangle TLC:

I) $\angle T=30^{\circ}=2* \angle C$

II) Side T is 3cm

Using I), we can find the measure of all 3 angles:

$\angle T= 30^{\circ}$

$\angle C= 15^{\circ}$

$\angle L = 180^{\circ}-30^{\circ}-15^{\circ}=135^{\circ}$

Next, use II) and the Law of sines to find the hypotenuse:

$\frac{3}{sin30}=\frac{h}{sin135}$

$h=3\sqrt{2}cm$

And we needed both statements to find it!

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Example Question #3 : Dsq: Calculating The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

3rd side triangle

For obtuse triangle ABC, what is the length of c?

(1) a=4 and b=7

(2) c is an integer, $10 \leq c <12$

Possible Answers:

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

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

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

EACH statement ALONE is sufficient.

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Correct answer:

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Explanation:

Since this is an obtuse triangle, pythagorean theorem does not apply.

Statement 1 by itself will only determine a range of values c utilizing the 3rd side rule of triangles. 3 < c < 11 . Therefore, statement 1 alone is insufficient.

Statement 2 by itself will determine that c is either 10 or 11. Therefore, statement 2 alone is insufficient.

When taken together, statements 1 and 2 define a definitive value for c: c=10 . Therefore, BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

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GMAT Math : Acute / Obtuse Triangles

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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

Example Question #12 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

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

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

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

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

Example Question #17 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Example Question #1 : Dsq: Calculating The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

Example Question #2 : Dsq: Calculating The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

Example Question #3 : Dsq: Calculating The Length Of The Hypotenuse Of An Acute / Obtuse Triangle

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