Squares, Rectangles, and Parallelograms - GED Math
Card 1 of 505
In Rhombus 
, 
. If 
is constructed, which of the following is true about 
?
In Rhombus
Tap to reveal answer
The figure referenced is below.
The sides of a rhombus are congruent by definition, so 
, making 
isosceles (and possibly equilateral).
Also, consecutive angles of a rhombus are supplementary, as they are with all parallelograms, so
.
, having measure greater than 
, is obtuse, making 
an obtuse triangle. Also, the triangle is not equilateral, since such a triangle must have three 
angles.
The correct response is that 
is obtuse and isosceles, but not equilateral.
The figure referenced is below.
The sides of a rhombus are congruent by definition, so
Also, consecutive angles of a rhombus are supplementary, as they are with all parallelograms, so
The correct response is that
← Didn't Know|Knew It →
Given Quadrilateral 
, which of these statements would prove that it is a parallelogram?
I) 
and 
II) 
and 
III) 
and 
are supplementary and 
and 
are supplementary
Given Quadrilateral
I)
II)
III)
Tap to reveal answer
Statement I asserts that two pairs of consecutive angles are congruent. This does not prove that the figure is a parallelogram. For example, an isosceles trapezoid has two pairs of congruent base angles, which are consecutive.
Statement II asserts that both pairs of opposite angles are congruent. By a theorem of geometry, this proves the quadrilateral to be a parallelogram.
Statement III asserts that two pairs of consecutive angles are supplementary. While all parallelograms have this characteristic, trapezoids do as well, so this does not prove the figure a parallelogram.
The correct response is Statement II only.
Statement I asserts that two pairs of consecutive angles are congruent. This does not prove that the figure is a parallelogram. For example, an isosceles trapezoid has two pairs of congruent base angles, which are consecutive.
Statement II asserts that both pairs of opposite angles are congruent. By a theorem of geometry, this proves the quadrilateral to be a parallelogram.
Statement III asserts that two pairs of consecutive angles are supplementary. While all parallelograms have this characteristic, trapezoids do as well, so this does not prove the figure a parallelogram.
The correct response is Statement II only.
← Didn't Know|Knew It →
You are given Parallelogram 
with 
. Which of the following statements, along with what you are given, would be enough to prove that Parallelogram 
is a rectangle?
I) 
II) 
III) 
You are given Parallelogram
I)
II)
III)
Tap to reveal answer
A rectangle is defined as a parallelogram with four right, or 
, angles.
Since opposite angles of a paralellogram are congruent, if one angle measures 
, so does its opposite. Since consecutive angles of a paralellogram are supplementary - that is, their degree measures total 
- if one angle measures 
, then both of the neighboring angles measure 
.
In short, in a parallelogram, if one angle is right, all are right and the parallelogram is a rectangle. All three statements assert that one angle is right, so from any one, it follows that the figure is a rectangle. The correct response is Statements I, II, or III.
Note that the sidelengths are irrelevant.
A rectangle is defined as a parallelogram with four right, or
Since opposite angles of a paralellogram are congruent, if one angle measures
In short, in a parallelogram, if one angle is right, all are right and the parallelogram is a rectangle. All three statements assert that one angle is right, so from any one, it follows that the figure is a rectangle. The correct response is Statements I, II, or III.
Note that the sidelengths are irrelevant.
← Didn't Know|Knew It →
If the rectangle has a width of 5 and a length of 10, what is the area of the rectangle?
If the rectangle has a width of 5 and a length of 10, what is the area of the rectangle?
Tap to reveal answer
Write the area for a rectangle.
Substitute the given dimensions.
The answer is: 
Write the area for a rectangle.
Substitute the given dimensions.
The answer is:
← Didn't Know|Knew It →
In the figure below, find the measure of the largest angle.
In the figure below, find the measure of the largest angle.
Tap to reveal answer
Recall that in a quadrilateral, the interior angles must add up to 
.
Thus, we can solve for 
:
Now, to find the largest angle, plug in the value of 
into each expression for each angle.
The largest angle is 
.
Recall that in a quadrilateral, the interior angles must add up to
Thus, we can solve for
Now, to find the largest angle, plug in the value of
The largest angle is
← Didn't Know|Knew It →
Use the following rectangle to answer the question:
Find the perimeter.
Use the following rectangle to answer the question:
Find the perimeter.
Tap to reveal answer
To find the perimeter of a rectangle, we will use the following formula:
where l is the length and w is the width of the rectangle.
Now, given the rectangle
we can see the length is 9cm and the width is 7cm. So, we can substitute. We get
To find the perimeter of a rectangle, we will use the following formula:
where l is the length and w is the width of the rectangle.
Now, given the rectangle
we can see the length is 9cm and the width is 7cm. So, we can substitute. We get
← Didn't Know|Knew It →
What is the perimeter of a square if the side length is 
?
What is the perimeter of a square if the side length is
Tap to reveal answer
A square has four congruent sides.
Multiply the quantity by four.
The answer is: 
A square has four congruent sides.
Multiply the quantity by four.
The answer is:
← Didn't Know|Knew It →
Note: Figure NOT drawn to scale
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown). The dirt path is six feet wide throughout. Which of the following polynomials gives the perimeter of the garden, in feet?
Note: Figure NOT drawn to scale
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown). The dirt path is six feet wide throughout. Which of the following polynomials gives the perimeter of the garden, in feet?
Tap to reveal answer
The length of the garden, in feet, is 
feet less than that of the entire lot, or
.
The width of the garden, in feet, is 
less than that of the entire lot, or
.
The perimeter, in feet, is twice the sum of the two:
The length of the garden, in feet, is
The width of the garden, in feet, is
The perimeter, in feet, is twice the sum of the two:
← Didn't Know|Knew It →
Refer to the above three figures. All parallel sides are so indicated.
Which of the figures can be called a quadrilateral?
Refer to the above three figures. All parallel sides are so indicated.
Which of the figures can be called a quadrilateral?
Tap to reveal answer
By definition, any polygon with four sides is called a quadrilateral. All three figures fit this description.
By definition, any polygon with four sides is called a quadrilateral. All three figures fit this description.
← Didn't Know|Knew It →
Refer to the above three figures. All parallel sides are so indicated.
Which of the figures can be called a parallelogram?
Refer to the above three figures. All parallel sides are so indicated.
Which of the figures can be called a parallelogram?
Tap to reveal answer
A parallelogram, by definition, has two pairs of parallel sides. Figures A and B fit that criterion, but Figure C does not.
A parallelogram, by definition, has two pairs of parallel sides. Figures A and B fit that criterion, but Figure C does not.
← Didn't Know|Knew It →
The length of each side of a square is increased by 12%. By what percent has its perimeter increased?
The length of each side of a square is increased by 12%. By what percent has its perimeter increased?
Tap to reveal answer
Let 
be the original sidelength of the square. Increasing this by 12% is the same as adding 0.12 times that sidelength to the original sidelength. The new sidelength is therefore
.
The perimeter of a square is four times its sidelength. Its original perimeter was 
; after the increase, it is 
. The percent of increase is therefore
.
Let
The perimeter of a square is four times its sidelength. Its original perimeter was
← Didn't Know|Knew It →
Note: Figure NOT drawn to scale
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown) six feet wide throughout. What is the perimeter of the garden?
Note: Figure NOT drawn to scale
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown) six feet wide throughout. What is the perimeter of the garden?
Tap to reveal answer
The inner rectangle, which represents the garden, has length and width 
feet and 
feet, respectively, so its perimeter is
feet.
The inner rectangle, which represents the garden, has length and width
← Didn't Know|Knew It →
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown). The dirt path is 
feet wide throughout. Which of the following polynomials gives the perimeter of the garden?
Note: Figure NOT drawn to scale.
Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in brown). The dirt path is
Tap to reveal answer
The length of the garden is 
than that of the entire lot, or
.
The width of the garden is 
than that of the entire lot, or
.
The perimeter is twice the sum of the two:
The length of the garden is
The width of the garden is
The perimeter is twice the sum of the two:
← Didn't Know|Knew It →
Refer to the above diagram. Parallel sides are so indicated.
Identify the above polygon.
Refer to the above diagram. Parallel sides are so indicated.
Identify the above polygon.
Tap to reveal answer
A four-sided figure, or quadrilateral, with one pair of parallel sides and its other sides nonparallel is called a trapezoid.
A four-sided figure, or quadrilateral, with one pair of parallel sides and its other sides nonparallel is called a trapezoid.
← Didn't Know|Knew It →
Note: Figure NOT drawn to scale
Give the ratio of the perimeter of Rectangle 
to that of Rectangle 
.
Note: Figure NOT drawn to scale
Give the ratio of the perimeter of Rectangle
Tap to reveal answer
The perimeter of Rectangle 
is
.
Opposite sides of a rectangle are congruent, so
and
.
The perimeter of Rectangle 
is
.
Opposite sides of a rectangle are congruent, so
,
,
and
.
The ratio of the perimeters is
- that is, 10 to 7.
The perimeter of Rectangle
Opposite sides of a rectangle are congruent, so
and
The perimeter of Rectangle
Opposite sides of a rectangle are congruent, so
and
The ratio of the perimeters is
← Didn't Know|Knew It →
Given Square 
, the distance from 
to 
is one mile. Which of the following gives the perimeter of the square, to the nearest foot?
Given Square
Tap to reveal answer
The segment 
is a diagonal of the square. It is also the hypotenuse of a right triangle whose legs both measure the same length, which we will call 
; this is also the length of one side of the square.
The right triangle formed is an isosceles triangle - and, subsequently, a 
triangle - and therefore,
is one mile, or 5,280 feet, long, so we substitute 
and solve for 
:
The perimeter, in feet, is four times this, or
feet.
The segment
The right triangle formed is an isosceles triangle - and, subsequently, a
The perimeter, in feet, is four times this, or
← Didn't Know|Knew It →
Which of the following can be the sidelengths of a rhombus?
Which of the following can be the sidelengths of a rhombus?
Tap to reveal answer
The four sides of a rhombus have equal length, so we can eliminate three choices by demonstrating that at least two sidelengths are not equal.
:
1,000 meters is, by definition, equal to 1 kilometer, not 0.1 kilometers. Therefore,
and this choice is incorrect.
:
1 mile is, by definition, equal to 5,280 feet, not 1,760 feet. Therefore,
and this choice is incorrect.
By definition, 1 decimeter, not 0.1 decimeter, is equal to 1 meter. Therefore,
and this choice is incorrect.
:
yard is equal to 
inches and, also, 
feet. Therefore,
All four sides have equal length so this is the rhombus. This is the correct choice.
The four sides of a rhombus have equal length, so we can eliminate three choices by demonstrating that at least two sidelengths are not equal.
1,000 meters is, by definition, equal to 1 kilometer, not 0.1 kilometers. Therefore,
and this choice is incorrect.
1 mile is, by definition, equal to 5,280 feet, not 1,760 feet. Therefore,
and this choice is incorrect.
By definition, 1 decimeter, not 0.1 decimeter, is equal to 1 meter. Therefore,
and this choice is incorrect.
All four sides have equal length so this is the rhombus. This is the correct choice.
← Didn't Know|Knew It →
Identify the above polygon.
Identify the above polygon.
Tap to reveal answer
A polygon with six sides is called a hexagon.
A polygon with six sides is called a hexagon.
← Didn't Know|Knew It →
Refer to the above figure. You are given that 
and that 
is acute.
Which of the following words accurately describes Polygon 
?
Refer to the above figure. You are given that
Which of the following words accurately describes Polygon
Tap to reveal answer
Polygon 
has four sides and is therefore a quadrilateral. 
, so 
. Also, since 
is acute and 
is right, 
, so 
.
The quadrilateral has one pair of parallel sides, and the other two sides are not parallel. Therefore, it is a trapezoid.
Polygon
The quadrilateral has one pair of parallel sides, and the other two sides are not parallel. Therefore, it is a trapezoid.
← Didn't Know|Knew It →
Note: Figure NOT drawn to scale.
Quadrilateral 
is a rhombus. Calculate its perimeter if:
Note: Figure NOT drawn to scale.
Quadrilateral
Tap to reveal answer
The four sides of a rhombus are congruent. Also, the diagonals of a rhombus are perpendicular bisectors to each other, so the four triangles they form are right triangles. Therefore, the Pythagorean theorem can be used to determine the common sidelength of Quadrilateral 
.
We focus on 
. The diagonals of a rhombus, as is the case with any parallelogram, are each the other's bisector, so
By the Pythagorean Theorem,
13 is the common length of the four sides of Quadrilateral 
, so its perimeter is 
.
The four sides of a rhombus are congruent. Also, the diagonals of a rhombus are perpendicular bisectors to each other, so the four triangles they form are right triangles. Therefore, the Pythagorean theorem can be used to determine the common sidelength of Quadrilateral
We focus on
By the Pythagorean Theorem,
13 is the common length of the four sides of Quadrilateral
← Didn't Know|Knew It →