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Intermediate Geometry Help » Plane Geometry » Quadrilaterals » Rhombuses
Example Question #1 : Rhombuses
Given: Rhombuses
and
.
True, false, or undetermined:Rhombus
Rhombus
.
Possible Answers:
False
Undetermined
True
Correct answer:
Undetermined
Explanation:
Two figures are congruentby definition if all of their corresponding sides are congruentand all of their corresponding angles are congruent.
By definition, a rhombus has four sides of equal length.If we let
be the common sidelength ofRhombus
and
be the common sidelength ofRhombus
, then, since
, it follows that
, so corresponding sides are congruent. However, no information is given about their angle measures. Therefore, it cannot be determined whether or not the two rhombuses are congruent.
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Example Question #2 : How To Find If Rhombuses Are Similar
Given: Rhombuses
and
.
and
True, false, or undetermined:Rhombus
Rhombus
.
Possible Answers:
True
False
Undetermined
Correct answer:
True
Explanation:
Two figures are similar by definition if all of their corresponding sides are proportional and all of their corresponding angles are congruent.
By definition, a rhombus has four sides that are congruent. If we let
be the common sidelength ofRhombus
and
be the common sidelength ofRhombus 
, it can easily be seen that the ratio of the length of each side of the former to that of the latter is the same ratio, namely,
.
Also, a rhombus being a parallelogram, its opposite angles are congruent, and its consecutiveangles are supplementary. Therefore, since
, it follows that
, and
. By a similar argument,
and
. Therefore,
Since all corresponding sides are proportional and all correspondingangles are congruent, it holds thatRhombus
Rhombus
.
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Example Question #1 : How To Find An Angle In A Rhombus
A rhombus has two interior angles with a measurement of
degrees. What is the measurement of each of the other two interior angles?
Possible Answers:
One angle is
degrees and the other angle must be
degrees.
Both of the remaining angles are
degrees.
All of the interior angles of the rhombus are
degrees.
Not enough information is provided to solve this problem.
Correct answer:
Both of the remaining angles aredegrees.
Explanation:
The four interior angles in any rhombus must have a sum of
degrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum of
degrees.
Thus, if a rhombus has two interior angles of
degrees, there must also be two angles that equal:
Check:
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Example Question #1 : How To Find An Angle In A Rhombus
Using the rhombus above, find the sum of angles
and
Possible Answers:
Correct answer:
Explanation:
The four interior angles in any rhombus must have a sum of
degrees.
The opposite interior angles must be equivalent, and the adjacent angles have a sum of
degrees.
Since
then,
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Example Question #3 : How To Find An Angle In A Rhombus
In the above rhombus, angle
degrees. Find the sum of angles 
and
Possible Answers:
Correct answer:
Explanation:
The four interior angles in any rhombus must have a sum ofdegrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum ofdegrees.
Since
, 
And
So,
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Example Question #1 : How To Find An Angle In A Rhombus
In the rhombus shown above, angle
has a measurement of
degrees. Find the measurement of angle
.
Possible Answers:
Correct answer:
Explanation:
The four interior angles in any rhombus must have a sum ofdegrees.
The opposite interior angles must be equivalent, and the adjacent angles have a sum ofdegrees.
Since angle
is adjacent to angle
, they must have a sum of
degrees.
The solution is:
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Example Question #5 : How To Find An Angle In A Rhombus
In the rhombus shown above, anglehas a measurement ofdegrees. Find the sum of angles
and
Possible Answers:
Correct answer:
Explanation:
The four interior angles in any rhombus must have a sum ofdegrees.
The opposite interior angles must be equivalent, and the adjacent angles have a sum ofdegrees.
Angles
and
are opposite interior angles, so they must have equivalent measurements.
The sum is:
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Example Question #1 : Quadrilaterals
Angle
has a measurement of
degrees. Find the sum of angles
and
Possible Answers:
Correct answer:
Explanation:
The four interior angles in any rhombus must have a sum ofdegrees.
The opposite interior angles must be equivalent, and the adjacent angles have a sum ofdegrees.
Since, both angles
and
are adjacent to angle
--find the measurement of one of these two angles by:
.
Angle
and angle
must each equal
degrees. So the sum of angles
and
degrees.
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Example Question #7 : How To Find An Angle In A Rhombus
Given: Parallelogram
such that
.
True or false:Parallelogram
cannot be a rhombus.
Possible Answers:
False
True
Correct answer:
False
Explanation:
A rhombus is defined to be a parallelogram with four congruent sides; there is no restriction as to the measures of the angles. Therefore, a rhombus can have angles of any measure. The correct choice is "false".
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Example Question #1 : How To Find An Angle In A Rhombus
Given: Rhombus
with diagonals
and
intersecting at point
.
True or false:
must bea right angle.
Possible Answers:
False
True
Correct answer:
True
Explanation:
One characteristic of a rhombusis that its diagonals are perpendicular. It follows thatmust be a right angle.
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