Acute / Obtuse Triangles - GMAT Quantitative

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Question

Which of the following cannot be the measure of a base angle of an isosceles triangle?

Answer

An isosceles triangle has two congruent angles by the Isosceles Triangle Theorem; these are the base angles. Since at least two angles of any triangle must be acute, a base angle must be acute - that is, it must measure under $90^{\circ }$ . The only choice that does not fit this criterion is $100 ^{\circ }$ , making this the correct choice.

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Question

Let the three interior angles of a triangle measure , and . Which of the following statements is true about the triangle?

Answer

If these are the measures of the interior angles of a triangle, then they total $180^{\circ }$ . Add the expressions, and solve for .

$x +\left ( x + 30 \right ) +\left ( 2x - 10 \right ) = 180$

One angle measures $40^{\circ }$ . The others measure:

$x + 30 = 40 + 30 = 70^{\circ }$

$2x-10 = 2 \cdot 40 - 10 = 80 - 10 = 70 ^{\circ }$

All three angles measure less than $90 ^{\circ }$ , so the triangle is acute. Also, there are two congruent angles, so by the converse of the Isosceles Triangle Theorem, two sides are congruent, and the triangle is isosceles.

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Question

Two angles of an isosceles triangle measure $(x + 20) ^{\circ}$ and $\left ( 100-x\right )^{ \circ}$ . What are the possible values of ?

Answer

In an isosceles triangle, at least two angles measure the same. Therefore, one of three things happens:

Case 1: The two given angles have the same measure.

$2x \div 2 =80 \div 2$

The angle measures are $60 ^{\circ}, 60 ^{\circ}, 60 ^{\circ}$ , making the triangle equianglular and, subsequently, equilateral. An equilateral triangle is considered isosceles, so this is a possible scenario.

Case 2: The third angle has measure $(x + 20) ^{\circ}$ .

Then, since the sum of the angle measures is 180,

$\left (x + 20 \right ) +\left (x +20 \right )+ \left ( 100-x \right ) = 180$

as before

Case 3: The third angle has measure $\left ( 100-x\right )^{ \circ}$

$\left (x + 20 \right ) +\left (100-x \right )+ \left ( 100-x \right ) = 180$

as before.

Thus, the only possible value of is 40.

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Question

Which of the following is true of a triangle with three angles whose measures have an arithmetic mean of $90^{\circ }$ ?

Answer

The sum of the measures of three angles of any triangle is 180; therefore, their mean is $180 \div 3 = 60$ , making a triangle with angles whose measures have mean 90 impossible.

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Question

Which of the following is true of a triangle with two $20 ^{\circ }$ angles?

Answer

The sum of the measures of three angles of any triangle is 180; therefore, if two angles have measure $20 ^{\circ }$ , the third must have measure $180 - 20 - 20 = 140^{\circ } > 90^{\circ }$ . This makes the triangle obtuse. Also, since the triangle has two congruent angles, it is isosceles by the Converse of the Isosceles Triangle Theorem.

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Question

Two angles of an isosceles triangle measure $\left (x + 30 \right )^{\circ }$ and $\left (x + 36 \right )^{\circ }$ . What are the possible value(s) of ?

Answer

In an isosceles triangle, at least two angles measure the same. Therefore, one of three things happens:

Case 1: The two given angles have the same measure.

This is a false statement, indicating that this situation is impossible.

Case 2: The third angle has measure $\left (x + 30 \right )^{\circ }$ .

Then, since the sum of the angle measures is 180,

$\left (x + 30 \right ) +\left (x + 30 \right )+ \left ( x + 36 \right ) = 180$

This makes the angle measures $58^{\circ },58^{\circ },64^{\circ }$ , a plausible scenario.

Case 3: the third angle has measure $\left (x + 36 \right )^{\circ }$

Then, since the sum of the angle measures is 180,

$\left (x + 30 \right ) +\left (x + 36 \right )+ \left ( x + 36 \right ) = 180$

This makes the angle measures $56^{\circ }, 62^{\circ }, 62^{\circ }$ , a plausible scenario.

Therefore, either or

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Question

$\Delta ABC \sim \Delta XYZ$

$m \angle A = 20 ^{\circ }; m \angle Y = 120 ^{\circ }$

Which of the following is true of $\Delta ABC$ ?

Answer

By similarity, $m \angle B = m \angle Y = 120 ^{\circ }$ .

Since measures of the interior angles of a triangle total $180 ^{\circ }$ ,

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

$20 + 120 + m \angle C = 180$

$140 + m \angle C = 180$

$140 + m \angle C -140 = 180-140$

$m \angle C =40^{\circ }$

Since the three angle measures of $\Delta ABC$ are all different, no two sides measure the same; the triangle is scalene. Also, since $m \angle B = 120 ^{\circ } > 90^{\circ }$ , the angle is obtuse, and $\Delta ABC$ is an obtuse triangle.

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Question

Two angles of a triangle measure $47 ^{\circ }$ and $23^{\circ }$ . What is the measure of the third angle?

Answer

The sum of the degree measures of the angles of a triangle is 180, so we can subtract the two angle measures from 180 to get the third:

$(180-47-23)^{\circ } = 110^{\circ }$

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Question

The angles of a triangle measure $x^{\circ }, x^{\circ },\left ( x -54 \right ) ^{\circ }$ . Evaluate .

Answer

The sum of the measures of the angles of a triangle total $180^{\circ }$ , so we can set up and solve for in the following equation:

$3x\div 3 =234 \div 3$

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Question

An exterior angle of $\Delta ABC$ with vertex measures $150^{\circ }$ ; an exterior angle of $\Delta ABC$ with vertex measures $110^{\circ }$ . Which is the following is true of $\Delta ABC$ ?

Answer

An interior angle of a triangle measures $180 ^{\circ }$ minus the degree measure of its exterior angle. Therefore:

$m \angle A = 180 ^{\circ } - 150 ^{\circ } = 30 ^{\circ }$

$m \angle B= 180 ^{\circ } - 110 ^{\circ } = 70 ^{\circ }$

The sum of the degree measures of the interior angles of a triangle is $180 ^{\circ }$ , so

$m \angle C = 180 ^{\circ } - (m \angle A + m \angle B) = 180 ^{\circ } - (30 ^{\circ }+ 70 ^{\circ }) = 180 ^{\circ } -100 ^{\circ } = 80 ^{\circ }$ .

Each angle is acute, so the triangle is acute; each angle is of a different measure, so the triangle has three sides of different measure, making it scalene.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

$m\angle DCF = 140 ^{\circ } , m \angle ABF = 134 ^{\circ }$

Evaluate $m \angle EFB$ .

Answer

The sum of the exterior angles of a triangle, one per vertex, is $360 ^{\circ}$ . $\angle DCF$ , $\angle ABF$ and $\angle EFB$ are exterior angles at different vertices, so

$m \angle EFB + m\angle DCF + m \angle ABF = 360 ^{\circ }$

$m \angle EFB + 140 ^{\circ } + 134 ^{\circ } = 360 ^{\circ }$

$m \angle EFB + 274 ^{\circ } = 360 ^{\circ }$

$m \angle EFB + 274 ^{\circ }- 274 ^{\circ } = 360 ^{\circ } - 274 ^{\circ }$

$m \angle EFB = 86^{\circ }$

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Question

In the following triangle:

The angle degrees

The angle degrees

(Figure not drawn on scale)

Find the value of .

Answer

Since , the following triangles are isoscele: .

If ADC, BDC, and BDA are all isoscele; then:

The angle degrees

The angle degrees, and

The angle degrees

Therefore:

The angle

The angle degrees, and

The angle

Since the sum of angles of a triangle is equal to 180 degrees then:

. So:

.

Now let us solve the equation for x:

$2x + 150 = 180 \rightarrow x = 15$

(See image below - not drawn on scale)

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Question

$\angle 1$ is an exterior angle of $\bigtriangleup ABC$ at $\angle B AC$ .

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

Statement 1: $\angle 1$ is acute.

Statement 2: $\angle B$ and $\angle C$ are both acute.

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 is insufficient. Every triangle has at least two acute angles, and Statement 2 only establishes that $\angle B$ and $\angle C$ are both acute; the third angle, $\angle B AC$ , can be acute, right, or obtuse, so the question of whether $\bigtriangleup ABC$ is an acute, right, or obtuse triangle is not settled.

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Question

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

Statement 1: $m \angle A + m \angle B < 90 ^{\circ }$

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

Answer

Assume Statement 1 alone. The sum of the measures of interior angles of a triangle is $180 ^{\circ }$ ;

$m \angle A + m \angle B < 90 ^{\circ }$ , or, equivalently, for some positive number ,

$m \angle A + m \angle B =( 90 - N) ^{\circ }$ ,

so

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

$( 90 - N) ^{\circ }+ m \angle C = 180 ^{\circ }$

$m \angle C = 180 ^{\circ } - ( 90 - N) ^{\circ } = ( 90+N) ^{\circ }$

Therefore, $m \angle C > 90 ^{\circ }$ , making $\angle C$ obtuse, and $\bigtriangleup ABC$ an obtuse triangle.

Assume Statement 2 alone. Since the sum of the squares of the lengths of two sides exceeds the square of the length of the third, it follows that $\bigtriangleup ABC$ is an obtuse triangle.

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Question

$\angle 1$ , $\angle 2$ , and $\angle 3$ are all exterior angles of $\bigtriangleup ABC$ with vertices , , and , respectively.

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

Statement 1: $\angle 1$ , $\angle 2$ , and $\angle 3$ are all obtuse angles.

Statement 2: $\angle A \cong \angle B$ .

Note: For purposes of this problem, $\angle A$ , $\angle B$ , and $\angle C$ will refer to the interior angles of the triangle at these vertices.

Answer

Assume Statement 1 alone. An exterior angle of a triangle forms a linear pair with the interior angle of the triangle of the same vertex. The two angles, whose measures total $180^{\circ }$ , must be two right angles or one acute angle and one obtuse angle. Since $\angle 1$ , $\angle 2$ , and $\angle 3$ are all obtuse angles, it follows that their respective interior angles - the three angles of $\bigtriangleup ABC$ - are all acute. This makes $\bigtriangleup ABC$ an acute triangle.

Statement 2 alone provides insufficient information to answer the question. For example, if $\angle A$ and $\angle B$ each measure $10^{\circ }$ and $\angle C$ measures $160^{\circ }$ , the sum of the angle measures is $180^{\circ }$ , $\angle A$ and $\angle B$ are congruent, and $\angle C$ is an obtuse angle (measuring more than $90^{\circ }$ ); this makes $\bigtriangleup ABC$ an obtuse triangle. But if $\angle A$ , $\angle B$ , and $\angle C$ each measure $60^{\circ }$ , the sum of the angle measures is again $180^{\circ }$ , $\angle A$ and $\angle B$ are again congruent, and all three angles are acute (measuring less than $90^{\circ }$ ); this makes $\bigtriangleup ABC$ an acute triangle.

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Question

The measures of the interior angles of a triangle are $x^{\circ }$ , $y^{\circ }$ , and $72^{\circ }$ . Also,

.

Evaluate .

Answer

The measures of the interior angles of a triangle have sum $180^{\circ }$ , so

Along with , a system of linear equations is formed that can be solved by adding:

$\underline{x+y = 108}$

$2y \div 2 = 86 \div 2$

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Question

The interior angles of a triangle have measures $x^{\circ }$ , $y^{\circ }$ , and $2x^{\circ }$ . Also,

.

Which of the following is closest to ?

Answer

The measures of the interior angles of a triangle have sum $180^{\circ }$ , so

, or

Along with , a system of linear equations is formed that can be solved by adding:

$\underline{x - y = 32}$

$4x \div 4 = 212 \div 4$

Of the given choices, 50 comes closest to the correct measure.

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Question

A triangle has interior angles whose measures are $x^{\circ }$ , $y^{\circ }$ , and $42 ^{\circ }$ . A second triangle has interior angles, two of whose measures are $\frac{1}{2}x^{\circ }$ and $\frac{1}{2}y^{\circ }$ . What is the measure of the third interior angle of the second triangle?

Answer

The measures of the interior angles of a triangle have sum $180^{\circ }$ , so

, or, equivalently,

$\frac{1}{2}\left (x+ y \right )= \frac{1}{2} \cdot 138$

$\frac{1}{2} x+ \frac{1}{2} y =69$

$\frac{1}{2}x^{\circ }$ and $\frac{1}{2}y^{\circ }$ are the measures of two interior angles of the second triangle, so if we let be the measure of the third angle, then

$\frac{1}{2} x+ \frac{1}{2} y +Z= 180$

By substitution,

and

.

The correct response is $111^{\circ }$ .

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Question

The measures of the interior angles of Triangle 1 are $x^{\circ }$ , $y^{\circ }$ , and $107^{\circ }$ . The measures of two of the interior angles of Triangle 2 are $2x^{\circ }$ and $2y^{\circ }$ . Which of the following is the measure of the third interior angle of Triangle 2?

Answer

The measures of the interior angles of a triangle have sum $180^{\circ }$ , so

, or, equivalently,

$2 (x+ y )= 2 \cdot 73$

$2x^{\circ }$ and $2y^{\circ }$ are the measures of two interior angles of the second triangle, so if we let be the measure of the third angle, then

By substitution,

The correct response is $34^{\circ }$ .

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Question

Triangle 1 has three interior angles with measures $x^{\circ }$ , $x^{\circ }$ , and $y^{\circ }$ . Triangle 1 has three interior angles with measures $\frac{3}{2} x^{\circ }$ , $\frac{3}{2} x^{\circ }$ , and $z^{\circ }$ .

Express in terms of .

Answer

The sum of the measures of the interior angles of a triangle is $180^{\circ }$ , so it can be determined from Triangle 1 that

$\frac{1}{2} \cdot 2x = \frac{1}{2} \cdot (180-y)$

$x =90-\frac{1}{2} y$

From Triangle 2, we can deduce that

$\frac{3}{2} x^{\circ } + \frac{3}{2} x^{\circ } + z = 180$

By substitution:

$z = 180 -3\left ( 90-\frac{1}{2} y \right )$

$z = 180 -270 +\frac{3}{2} y$

$z = \frac{3}{2} y - 90$

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