Plane Geometry - GRE Quantitative Reasoning
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Quantitative Comparison
Quantity A: The degree measure of any angle in an equilateral triangle
Quantity B: The degree measure of any angle in a regular hexagon
Quantitative Comparison
Quantity A: The degree measure of any angle in an equilateral triangle
Quantity B: The degree measure of any angle in a regular hexagon
We know the three angles in a triangle add up to 180 degrees, and all three angles are 60 degrees in an equilateral triangle.
A hexagon has six sides, and we can use the formula degrees = (# of sides – 2) * 180. Then degrees = (6 – 2) * 180 = 720 degrees. Each angle is 720/6 = 120 degrees.
Quantity B is greater.
We know the three angles in a triangle add up to 180 degrees, and all three angles are 60 degrees in an equilateral triangle.
A hexagon has six sides, and we can use the formula degrees = (# of sides – 2) * 180. Then degrees = (6 – 2) * 180 = 720 degrees. Each angle is 720/6 = 120 degrees.
Quantity B is greater.
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Quantity A: Double the measure of a single interior angle of an equilateral triangle.
Quantity B: The measure of a single interior angle of a hexagon.
Quantity A: Double the measure of a single interior angle of an equilateral triangle.
Quantity B: The measure of a single interior angle of a hexagon.
Begin with Quantity A. We know the measure of one angle in an equilateral triangle is 60. Therefore, double the angle is 120 degrees.
For the hexagon, use the formula for the sum of the interior angles:
where n= number of sides in a regular polygon
If the sum of the interior angles of a regular hexagon is 
degrees, then one angle is 
degrees.
The two quantities are equal.
Begin with Quantity A. We know the measure of one angle in an equilateral triangle is 60. Therefore, double the angle is 120 degrees.
For the hexagon, use the formula for the sum of the interior angles:
where n= number of sides in a regular polygon
If the sum of the interior angles of a regular hexagon is
The two quantities are equal.
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In a given parallelogram, the measure of one of the interior angles is 25 degrees less than another. What is the approximate measure rounded to the nearest degree of the larger angle?
In a given parallelogram, the measure of one of the interior angles is 25 degrees less than another. What is the approximate measure rounded to the nearest degree of the larger angle?
There are two components to solving this geometry puzzle. First, one must be aware that the sum of the measures of the interior angles of a parallelogram is 360 degrees (sum of the interior angles of a figure = 180(n-2), where n is the number of sides of the figure). Second, one must know that the other two interior angles are doubles of those given here. Thus if we assign one interior angle as x and the other as x-25, we find that x + x + (x-25) + (x-25) = 360. Combining like terms leads to the equation 4x-50=360. Solving for x we find that x = 410/4, 102.5, or approximately 103 degrees. Since x is the measure of the larger angle, this is our answer.
There are two components to solving this geometry puzzle. First, one must be aware that the sum of the measures of the interior angles of a parallelogram is 360 degrees (sum of the interior angles of a figure = 180(n-2), where n is the number of sides of the figure). Second, one must know that the other two interior angles are doubles of those given here. Thus if we assign one interior angle as x and the other as x-25, we find that x + x + (x-25) + (x-25) = 360. Combining like terms leads to the equation 4x-50=360. Solving for x we find that x = 410/4, 102.5, or approximately 103 degrees. Since x is the measure of the larger angle, this is our answer.
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Figure 
is a parallelogram.
What is 
in the figure above?
Figure
What is
Because of the character of parallelograms, we know that our figure can be redrawn as follows:
Because it is a four-sided figure, we know that the sum of the angles must be 
. Thus, we know:
Solving for 
, we get:
Because of the character of parallelograms, we know that our figure can be redrawn as follows:
Because it is a four-sided figure, we know that the sum of the angles must be
Solving for
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Figure 
is a parallelogram.
Quantity A: The largest angle of 
.
Quantity B: 
Which of the following is true?
Figure
Quantity A: The largest angle of
Quantity B:
Which of the following is true?
By using the properties of parallelograms along with those of supplementary angles, we can rewrite our figure as follows:
Recall, for example, that angle 
is equal to:
, hence 
Now, you know that these angles can all be added up to 
. You should also know that 
Therefore, you can write:
Simplifying, you get:
Now, this means that:
and 
. Thus, the two values are equal.
By using the properties of parallelograms along with those of supplementary angles, we can rewrite our figure as follows:
Recall, for example, that angle
Now, you know that these angles can all be added up to
Therefore, you can write:
Simplifying, you get:
Now, this means that:
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In a five-sided polygon, one angle measures 
. What are the possible measurements of the other angles?
In a five-sided polygon, one angle measures
To find the sum of the interior angles of any polygon, use the formula 
, where n represents the number of sides of a polygon.
In this case:
The sum of the interior angles will be 540. Go through each answer choice and see which one adds up to 540 (including the original angle given in the problem).
The only one that does is 120, 115, 95, 105 and the original angle of 105.
To find the sum of the interior angles of any polygon, use the formula
In this case:
The sum of the interior angles will be 540. Go through each answer choice and see which one adds up to 540 (including the original angle given in the problem).
The only one that does is 120, 115, 95, 105 and the original angle of 105.
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In a particular heptagon (a seven-sided polygon) the sum of four equal interior angles, each equal to 
degrees, is equivalent to the sum of the remaining three interior angles.
Quantity A: 
Quantity B: 
In a particular heptagon (a seven-sided polygon) the sum of four equal interior angles, each equal to
Quantity A:
Quantity B:
The sum of interior angles in a heptagon is 
degrees. Note that to find the sum of interior angles of any polygon, it is given by the formula:
degrees, where 
is the number of sides of the polygon.
Three interior angles (call them 
) are unknown, but we are told that the sum of them is equal to the sum of four other equivalent angles (which we'll designate 
):
Further more, all of these angles must sum up to 
degrees:
We may not be able to find 
, 
, or 
, indvidually, but the problem does not call for that, and we need only use their relation to 
, as stated in the first equation with them. Utilizing this in the second, we find:
The sum of interior angles in a heptagon is
Three interior angles (call them
Further more, all of these angles must sum up to
We may not be able to find
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What is the value of 
in the figure above?
What is the value of
Always begin working through problems like this by filling in all available information. We know that we can fill in two of the angles, giving us the following figure:
Now, we know that for any polygon, the total number of degrees in the figure can be calculated by the equation:
, where 
is the number of sides.
Thus, for our figure, we have:
Based on this, we know:
Simplifying, we get:
Solving for 
, we get:
or 
Always begin working through problems like this by filling in all available information. We know that we can fill in two of the angles, giving us the following figure:
Now, we know that for any polygon, the total number of degrees in the figure can be calculated by the equation:
Thus, for our figure, we have:
Based on this, we know:
Simplifying, we get:
Solving for
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Quantity A: The measure of the largest angle in the figure above.
Quantity B: 
Which of the following is true?
Quantity A: The measure of the largest angle in the figure above.
Quantity B:
Which of the following is true?
To begin, recall that the total degrees in any figure can be calculated by:
, where 
represents the total number of sides. Thus, we know for our figure that:
Now, based on our figure, we can make the equation:
Simplifying, we get:
or 
This means that 
is 
. Quantity A is larger.
To begin, recall that the total degrees in any figure can be calculated by:
Now, based on our figure, we can make the equation:
Simplifying, we get:
This means that
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A triangle has three internal angles of 75, 60, and x. What is x?
A triangle has three internal angles of 75, 60, and x. What is x?
The internal angles of a triangle must add up to 180. 180 - 75 -60= 45.
The internal angles of a triangle must add up to 180. 180 - 75 -60= 45.
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An isosceles triangle has an angle of 110°. Which of the following angles could also be in the triangle?
An isosceles triangle has an angle of 110°. Which of the following angles could also be in the triangle?
An isosceles triangle always has two equal angles. As there cannot be another 110° (the triangle cannot have over 180° total), the other two angles must equal eachother. 180° - 110° = 70°. 70° represents the other two angles, so it needs to be divided in 2 to get the answer of 35°.
An isosceles triangle always has two equal angles. As there cannot be another 110° (the triangle cannot have over 180° total), the other two angles must equal eachother. 180° - 110° = 70°. 70° represents the other two angles, so it needs to be divided in 2 to get the answer of 35°.
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An isosceles triangle ABC is laid flat on its base. Given that <B, located in the lower left corner, is 84 degrees, what is the measurement of the top angle, <A?
An isosceles triangle ABC is laid flat on its base. Given that <B, located in the lower left corner, is 84 degrees, what is the measurement of the top angle, <A?
Since the triangle is isosceles, and <A is located at the top of the triangle that is on its base, <B and <C are equivalent. Since <B is 84 degrees, <C is also. There are 180 degrees in a triangle so 180 - 84 - 84 = 12 degrees.
Since the triangle is isosceles, and <A is located at the top of the triangle that is on its base, <B and <C are equivalent. Since <B is 84 degrees, <C is also. There are 180 degrees in a triangle so 180 - 84 - 84 = 12 degrees.
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Triangle ABC is isosceles
x and y are positive integers
A
---
x
B
---
y
Triangle ABC is isosceles
x and y are positive integers
A
---
x
B
---
y
Since we are given expressions for the two congruent angles of the isosceles triangle, we can set the expressions equal to see how x relates to y. We get,
x – 3 = y – 7 --> y = x + 4
Logically, y must be the greater number if it takes x an additional 4 units to reach its value (knowing they are both positive integers).
Since we are given expressions for the two congruent angles of the isosceles triangle, we can set the expressions equal to see how x relates to y. We get,
x – 3 = y – 7 --> y = x + 4
Logically, y must be the greater number if it takes x an additional 4 units to reach its value (knowing they are both positive integers).
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An isosceles triangle has one obtuse angle that is 
. What is the value of one of the other angles?
An isosceles triangle has one obtuse angle that is
We know that an isosceles triangel has two equal sides and thus two equal angles opposite those equal sides. Because there is one obtuse angle of 112 degrees we automatically know that this angle is the vertex. If you sum any triangle's interior angles, you always get 180 degrees.
180 – 112 = 68 degrees. Thus there are 68 degrees left for the two equal angles. Each angle must therefore measure 34 degrees.
We know that an isosceles triangel has two equal sides and thus two equal angles opposite those equal sides. Because there is one obtuse angle of 112 degrees we automatically know that this angle is the vertex. If you sum any triangle's interior angles, you always get 180 degrees.
180 – 112 = 68 degrees. Thus there are 68 degrees left for the two equal angles. Each angle must therefore measure 34 degrees.
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In the figure above, what is the value of angle x?
In the figure above, what is the value of angle x?
To find the top inner angle, recognize that a straight line contains 180o; thus we can subtract 180 – 115 = 65o. Since we are given the other interior angle of 30 degrees, we can add the two we know: 65 + 30 = 95o.
180 - 95 = 85
To find the top inner angle, recognize that a straight line contains 180o; thus we can subtract 180 – 115 = 65o. Since we are given the other interior angle of 30 degrees, we can add the two we know: 65 + 30 = 95o.
180 - 95 = 85
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The three angles in a triangle measure 3_x_, 4_x_ + 10, and 8_x_ + 20. What is x?
The three angles in a triangle measure 3_x_, 4_x_ + 10, and 8_x_ + 20. What is x?
We know the angles in a triangle must add up to 180, so we can solve for x.
3_x_ + 4_x_ + 10 + 8_x_ + 20 = 180
15_x_ + 30 = 180
15_x_ = 150
x = 10
We know the angles in a triangle must add up to 180, so we can solve for x.
3_x_ + 4_x_ + 10 + 8_x_ + 20 = 180
15_x_ + 30 = 180
15_x_ = 150
x = 10
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In triangle ABC, AB=6, AC=3, and BC=4.
Quantity A Quantity B
angle C the sum of angle A and angle B
In triangle ABC, AB=6, AC=3, and BC=4.
Quantity A Quantity B
angle C the sum of angle A and angle B
The given triangle is obtuse. Thus, angle 
is greater than 90 degrees. A triangle has a sum of 180 degrees, so angle 
+ angle 
+ angle 
= 180. Since angle C is greater than 90 then angle 
+ angle 
must be less than 90 and it follows that Quantity A is greater.
The given triangle is obtuse. Thus, angle
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What is the circumference of a circle with an area of 36π?
What is the circumference of a circle with an area of 36π?
We know that the area of a circle can be expressed: a = πr2
If we know that the area is 36π, we can substitute this into said equation and get: 36π = πr2
Solving for r, we get: 36 = r2; (after taking the square root of both sides:) 6 = r
Now, we know that the circuference of a circle is expressed: c = πd. Since we know that d = 2r (two radii, placed one after the other, make a diameter), we can rewrite the circumference equation to be: c = 2πr
Since we have r, we can rewrite this to be: c = 2π*6 = 12π
We know that the area of a circle can be expressed: a = πr2
If we know that the area is 36π, we can substitute this into said equation and get: 36π = πr2
Solving for r, we get: 36 = r2; (after taking the square root of both sides:) 6 = r
Now, we know that the circuference of a circle is expressed: c = πd. Since we know that d = 2r (two radii, placed one after the other, make a diameter), we can rewrite the circumference equation to be: c = 2πr
Since we have r, we can rewrite this to be: c = 2π*6 = 12π
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Circle A has an area of 
. What is the perimeter of an enclosed semi-circle with half the radius of circle A?
Circle A has an area of
Based on our information, we know that the 121π = πr2; 121 = r2; r = 11.
Our other circle with half the radius of A has a diameter equal to the radius of A. Therefore, the circumference of this circle is 11π. Half of this is 5.5π. However, since this is a semi circle, it is enclosed and looks like this:
Therefore, we have to include the diameter in the perimeter. Therefore, the total perimeter of the semi-circle is 5.5π + 11.
Based on our information, we know that the 121π = πr2; 121 = r2; r = 11.
Our other circle with half the radius of A has a diameter equal to the radius of A. Therefore, the circumference of this circle is 11π. Half of this is 5.5π. However, since this is a semi circle, it is enclosed and looks like this:
Therefore, we have to include the diameter in the perimeter. Therefore, the total perimeter of the semi-circle is 5.5π + 11.
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Which is greater: the circumference of a circle with an area of 
, or the perimeter of a square with side length 
inches?
Which is greater: the circumference of a circle with an area of
Starting with the circle, we need to find the radius in order to get the circumference. Find 
by plugging our given area into the equation for the area of a circle:
Then calculate circumference:
(approximating 
as 3.14)
To find the perimeter of the square, we can use 
, where 
is the perimeter and 
is the side length:
, so the circle's circumference is greater.
Starting with the circle, we need to find the radius in order to get the circumference. Find
Then calculate circumference:
To find the perimeter of the square, we can use
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