Acute / Obtuse Triangles - GMAT Quantitative

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Question

Find the hypotenuse of an obtuse triangle.

Statement 1: Two given lengths with an inscribed angle.

Statement 2: Two known angles.

Answer

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 $\textup{c}$ .

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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Question

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

(1) and

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

Answer

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

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Question

$\measuredangle A=50, \measuredangle B=60$ . What is the measure of c?

(1)

(2)

Answer

Since , , , therefore, thus making this an acute triangle. Pythagorean theorem will not apply.

With the information in statement 1, we can't determine the lengths of any other sides. Therefore, Statement 1 alone is not sufficient.

With the information in statement 2, we can't determine the lengths of any other sides. Therefore, Statement 2 alone is not sufficient.

Using the Third Side Rule for triangles, the information in statements 1 and 2 together would allow us to determine the range of values for c. , but this does not provide a definitive value for c. Therefore, Both statements together are not sufficient.

Therefore - the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.

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Question

Find the length of the hypotenuse of obtuse triangle TLC:

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

II) Side T is

Answer

Find the length of the hypotenuse of obtuse triangle TLC:

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

II) Side T is

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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Question

and

Is it true that $\Delta ABC \cong \Delta DEF$ ?

$\angle B \cong \angle E$

$\angle C \cong \angle F$

Answer

If only one of the statements is known to be true, the only congruent pairs that are known between the triangles comprise two sides and a non-included angle; this information cannot prove congruence between the triangles. If both are known to be true, however, they, along with either of the given side congruences, set up the conditions for the Angle-Angle-Side Theorem, and the triangles can be proved congruent.

The answer is that both statements together are sufficient to answer the question, but not either alone.

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Question

You are given two triangles $\Delta ABC$ and $\Delta DEF$ ; with $\overline{AB} \cong \overline{DE}$ and $\overline{BC} \cong \overline{EF}$ . Which side is longer, $\overline{AC}$ or $\overline{DF}$ ?

Statement 1: $m \angle B > m \angle E$

Statement 2: $\angle B$ and $\angle F$ are both right angles.

Answer

We are given two triangles with two side congruences between them. If we compare their included angles (the angles that they form), the angle that is of greater measure will have the longer side opposite it. This is known as the Hinge Theorem.

The first statement says explicitly that the first included angle, $\angle B$ , has greater measure than the second, $\angle E$ , so the side opposite $\angle B$ , $\overline{AC}$ , has greater measure than $\overline{DF}$ .

The second statement is not so explicit. But if $\angle F$ is a right angle, $\angle E$ must be acute, and if $\angle B$ is right, then $m \angle B > m \angle E$ , which again proves that .

The answer is that either statement alone is sufficient to answer the question.

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Question

and

Is it true that $\Delta ABC \cong \Delta DEF$ ?

$\angle B \cong \angle E$

$\angle C \cong \angle F$

Answer

If only one of the statements is known to be true, the only congruent pairs that are known between the triangles comprise two sides and a non-included angle; this information cannot prove congruence between the triangles. If both are known to be true, however, they, along with either of the given side congruences, set up the conditions for the Angle-Angle-Side Theorem, and the triangles can be proved congruent.

The answer is that both statements together are sufficient to answer the question, but not either alone.

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Question

You are given two triangles $\Delta ABC$ and $\Delta DEF$ ; with $\overline{AB} \cong \overline{DE}$ and $\overline{BC} \cong \overline{EF}$ . Which side is longer, $\overline{AC}$ or $\overline{DF}$ ?

Statement 1: $m \angle B > m \angle E$

Statement 2: $\angle B$ and $\angle F$ are both right angles.

Answer

We are given two triangles with two side congruences between them. If we compare their included angles (the angles that they form), the angle that is of greater measure will have the longer side opposite it. This is known as the Hinge Theorem.

The first statement says explicitly that the first included angle, $\angle B$ , has greater measure than the second, $\angle E$ , so the side opposite $\angle B$ , $\overline{AC}$ , has greater measure than $\overline{DF}$ .

The second statement is not so explicit. But if $\angle F$ is a right angle, $\angle E$ must be acute, and if $\angle B$ is right, then $m \angle B > m \angle E$ , which again proves that .

The answer is that either statement alone is sufficient to answer the question.

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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

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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