Triangles - GMAT Quantitative Reasoning

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Question

Right triangle has length $AC=\sqrt{3}$ and . How many degrees is angle ?

Answer

For any right triangle, whose sides are in ratio $n, \sqrt{3}n, 2n$ , where is a constant, its angles must be $60^{\circ}$ and $30^{\circ}$ . Here the triangle has its sides in that ratio with $n=\sqrt{3}$ . Therefore, angle B must be the smallest angle, $30^{\circ}$ , since it is the angle between the two longest sides. This rule is really useful on the test, and it is advised to memorize it!

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Question

Of the two acute angles of a right triangle, one measures fifteen degrees less than twice the other. What is the measure of the smaller of the two angles?

Answer

Let one of the angles measure ; then the other angle measures . The sum of the measures of the acute angles of a triangle is $90^{\circ }$ , so we can set up and solve this equation:

$2x - 15 = 2 \cdot 30 - 15 = 55 ^{\circ }$

The acute angles measure $35^{\circ },55 ^{\circ }$ ; since we want the smaller of the two, $35^{\circ }$ is the correct choice.

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Question

For a certain right triangle, the angle between the base and the hypotenuse is 36 degrees. What is the measure of the only remaining unknown angle?

Answer

By definition, one of the angles of a right triangle must be 90 degrees. We are given the measure of another angle in the problem, so we now know the measure of two angles in the triangle. Because the sum of the angles of any triangle is 180 degrees, we can then solve for the measure of the only remaining unknown angle:

$90^{\circ}+36^{\circ}+x=180^{\circ}$

$x=180^{\circ}-90^{\circ}-36^{\circ}$

$x=54^{\circ}$

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Question

Triangle is a right triangle, with . What is the size of angle $\widehat{BCA}$ ?

Answer

Triangle ABC is an isosceles right triangle. Therefore, its angles at the basis BC will always be $45^{\circ}$ .

This stems from the fact that the sums of the angles of a triangle are $180^{\circ}$ and in our case with ABC a right and isosceles triangle, $180-90=90^{\circ}$ , therefore for the two remaining angles are equal.

There are 90 degrees left, therefore to find the measure of each angle we do the following,

$\frac{90}{2}=45^{\circ}$ .

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Question

is a right triangle, with sides $AB=3, AC=\sqrt{3}$ . What is the size of angle $\widehat{ACB}$ ?

Answer

Here, we can tell the size of the angles by recognizing the length of the sides indicative of a right triangle with angles $30^{\circ}-60^{\circ}-90^{\circ}$ .

Indeed, the length of the sides are $\sqrt{3}$ and . Any triangle with sides $n,n\sqrt{3},2n$ , where is a constant, will have angles $30^{\circ}-60^{\circ}-90^{\circ}$ .

In our case $n=\sqrt3$ . Therefore, angle $\widehat{CBA}$ will be the smallest of the three possible angles, since it is between the two longest sides ( the hypotenuse and AB, which is longer than AC). Therefore the larger angle $\widehat{ACB}$ will be $60^\circ$ thus arriving at our final answer.

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Question

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

.

Evaluate .

Answer

The measures of the acute angles of a right triangle have sum $90^{\circ }$ , so

Along with , a system of linear equations can be formed and solved as follows:

$\underline{x+y = 90}$

$2x \div 2 = 105 \div 2$

$x = 52 \frac{1}{2}$

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Question

Given a right triangle $\bigtriangleup ABC$ with right angle $\angle B$ , what is the measure of $\angle A$ ?

Statement 1: $m \angle A - m \angle C = 10^{\circ }$

Statement 2: $4 \cdot m \angle A = 5 \cdot m \angle C$

Answer

Let be the measure of $\angle A$ . The sum of the measures of the acute angles of a right triangle, $\angle A$ and $\angle C$ , is $90^{\circ }$ , so

$m \angle A + m \angle C = 90^{\circ }$

$X + m \angle C = 90^{\circ }$

$X + m \angle C -X = 90^{\circ } -X$

$m \angle C = 90^{\circ } -X$

Assume Statement 1 alone. This can be rewritten:

$m \angle A - m \angle C = 10^{\circ }$

$X - (90^{\circ } -X) = 10^{\circ }$

$2 X - 90^{\circ }= 10^{\circ }$

$2 X - 90^{\circ }+ 90^{\circ } = 10^{\circ } + 90^{\circ }$

$2 X = 10 0^{\circ }$

$2 X \div 2 = 10 0^{\circ } \div 2$

$X = 50^{\circ }$

Assume Statement 2 alone. This can be rewritten:

$4 \cdot m \angle A = 5 \cdot m \angle C$

$4 X = 5 (90^{\circ }-X)$

$4 X = 450 ^{\circ }-5X$

$4 X+ 5X = 450 ^{\circ }-5X + 5X$

$9X = 450 ^{\circ }$

$9X\div 9 = 450 ^{\circ } \div 9$

$X = 50^{\circ }$

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Question

A right triangle $\bigtriangleup ABC$ with right angle $\angle B$ ; all of its interior angles have degree measures that are whole numbers. What is the measure of $\angle A$ ?

Statement 1: $m \angle A$ is a multiple of 2 and 7.

Statement 2: $m \angle A$ is a multiple of 3 and 4.

Answer

An acute angle must have measure less than $90^{\circ }$ .

Assume Statement 1 alone. We are looking for a whole number that is a multiple of both 2 and 7, and is less than 90. There are several such numbers - 14 and 28, for example. There is no way of eliminating any of them, so the question is left unanswered.

Statement 2 alone provides insufficient information for similar reasons, since there are several whole numbers less than 90 that are multiples of 3 and 4 - 12 and 24, for example - with no way of eliminating any of them.

Now assume both statements to be true. 3, 4, and 7 are relatively prime - the greatest common factor of the four is 1 - so in order to find the least common multiple of the four, we need to multiply them. This product is

$3 \cdot 4 \cdot 7 = 84$ ,

which is also a multiple of 2. Any other multiple of all four numbers must be a multiple of 84, but any other positive multiple of 84 is greater than 90. Therefore, from the two statements together, it can be deduced that $\angle A$ has measure $84^{\circ }$ .

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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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