Acute / Obtuse Triangles - GMAT Quantitative

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Question

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) $m=k+4$

Answer

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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Question

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

(1)

(2)

Answer

There is an implied condition: . 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 into . Now we have , which means .

For statement (2), we can rewrite the equation to be and then plug into , making it

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

Then we can solve for and get .

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Question

True or false: The perimeter of $\bigtriangleup ABC$ is greater than 50.

Statement 1: $\bigtriangleup ABC$ is an isosceles triangle.

Statement 2: and

Answer

Statement 1 is insufficient, as it only gives that two sides are of equal length; it gives no side lengths, nor does it give any measurements that yield the side lengths.

Assume Statement 2 alone. By the Triangle Inequality, the length of each side must be less than the sum of the lengths of the other two, so

We can find the minimum of the perimeter:

$p =\left (AB+ AC \right ) + BC > 28+28 = 56 > 50$ :

The perimeter of $\bigtriangleup ABC$ is greater than 50.

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Question

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.

Answer

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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Question

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.

Answer

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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Question

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

Answer

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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Question

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

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

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

Answer

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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Question

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

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

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

Answer

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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Question

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.

Answer

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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Question

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

Answer

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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Question

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

Answer

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 $m \widehat{AB} = m \widehat{BC} =m \widehat{AC} =120 ^{\circ }$ .

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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Question

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.

Answer

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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Question

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.

Answer

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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Question

Is $\Delta ABC$ an isosceles triangle?

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

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

Answer

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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Question

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

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

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

Answer

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 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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Question

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

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

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

Answer

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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Question

$\angle 1$ 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: $m \angle B + m \angle C = 80 ^{\circ }$

Answer

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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Question

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

Statement 1:

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

Answer

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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Question

Which of two triangles has greater area, $\Delta ABC$ or $\Delta DEF$ ?

Statement 1: $\angle A \cong \angle D; \angle B \cong \angle E$

Statement 2: $AB = 1.1 \cdot DE$

Answer

Statement 1 alone proves that $\Delta ABC \sim \Delta DEF$ by the Angle-Angle Postulate, but does not prove anything about the sides, which would be needed to answer this question. Statement 2 alone only gives a relationship between one side of each triangle; without any further information, this is insufficient.

The two statements together, however, present sufficient evidence. Statement 1 proves that the triangles are similar. Statement 2 gives the ratio of one side in $\Delta ABC$ to the corresponding side in $\Delta DEF$ , so, as the triangles are similar, this ratio is shared by all three pairs of corresponding sides. Since

$\frac{AB }{DE}= 1.1$ ,

1.1 is this common ratio, and the ratio of the area of $\Delta ABC$ to $\Delta DEF$ is $1.1^{2} = 1.21$ , making $\Delta ABC$ the larger triangle.

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Question

True or false:

$\Delta ABC \sim \Delta DE F$

Statement 1: $\frac{AB }{DE}= \frac{BC }{EF}$

Statement 2: $\angle A \cong \angle D$ and $\angle B \cong \angle E$

Answer

Statement 1 alone gives a proportion between two pairs of corresponding sides of the triangles. This is not enough to prove the triangles similar without the third side proportionality (SSS Simiilarity statement) or the congruence of the included angles (SAS Similarity statement).

Statement 2 gives two congruencies between corresponding angles, which by the Angle-Angle statement is enough to prove the triangles similar.

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