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Example Questions
Example Question #1 : Rhombuses
Given: Rhombuses and
.
True, false, or undetermined: Rhombus Rhombus
.
False
Undetermined
True
Undetermined
Two figures are congruent by definition if all of their corresponding sides are congruent and all of their corresponding angles are congruent.
By definition, a rhombus has four sides of equal length. If we let be the common sidelength of Rhombus
and
be the common sidelength of Rhombus
, 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.
Example Question #2 : How To Find If Rhombuses Are Similar
Given: Rhombuses and
.
and
True, false, or undetermined: Rhombus Rhombus
.
True
False
Undetermined
True
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 of Rhombus
and
be the common sidelength of Rhombus
, 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 consecutive angles are supplementary. Therefore, since , it follows that
, and
. By a similar argument,
and
. Therefore,
Since all corresponding sides are proportional and all corresponding angles are congruent, it holds that Rhombus Rhombus
.
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?
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.
Both of the remaining angles are degrees.
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:
Example Question #1 : How To Find An Angle In A Rhombus
Using the rhombus above, find the sum of angles and
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,
Example Question #3 : How To Find An Angle In A Rhombus
In the above rhombus, angle degrees. Find the sum of angles
and
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 ,
And
So,
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
.
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 angle is adjacent to angle
, they must have a sum of
degrees.
The solution is:
Example Question #5 : How To Find An Angle In A Rhombus
In the rhombus shown above, angle has a measurement of
degrees. Find the sum of angles
and
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.
Angles and
are opposite interior angles, so they must have equivalent measurements.
The sum is:
Example Question #1 : Quadrilaterals
Angle has a measurement of
degrees. Find the sum of angles
and
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, 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.
Example Question #1 : How To Find An Angle In A Rhombus
Given: Parallelogram such that
.
True or false: Parallelogram cannot be a rhombus.
True
False
False
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".
Example Question #1 : How To Find An Angle In A Rhombus
Given: Rhombus with diagonals
and
intersecting at point
.
True or false: must be a right angle.
True
False
True
One characteristic of a rhombus is that its diagonals are perpendicular. It follows that must be a right angle.
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