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

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

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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

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