Rhombuses - Intermediate Geometry

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Question

A rhombus has two interior angles with a measurement of degrees. What is the measurement of each of the other two interior angles?

Answer

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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Question

In the above rhombus, angle degrees. Find the sum of angles and

Answer

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 $\measuredangle C=79^\circ$ , $\measuredangle B =180-79=101^\circ$

And $\measuredangle B=\measuredangle D$

So, $101+101=202^\circ$

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Question

Using the rhombus above, find the sum of angles and

Answer

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 $\measuredangle C=79^\circ$ then, $\measuredangle A=79^\circ$

$79+79=158^\circ$

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Question

In the rhombus shown above, angle has a measurement of degrees. Find the measurement of angle .

Answer

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:

$180-63=117^\circ$

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Question

Angle has a measurement of degrees. Find the sum of angles and

Answer

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.

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Question

In the rhombus shown above, angle has a measurement of degrees. Find the sum of angles and

Answer

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:

$63+63=126^\circ$

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Question

Given: Parallelogram such that $m \angle A = 80^{\circ }$ .

True or false: Parallelogram cannot be a rhombus.

Answer

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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Question

Given: Rhombus with diagonals $\overline{AC}$ and $\overline{BD}$ intersecting at point .

True or false: $\angle AXB$ must be a right angle.

Answer

One characteristic of a rhombus is that its diagonals are perpendicular. It follows that $\angle AXB$ must be a right angle.

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Question

Given Rhombus and diagonal $\overline{ BD}$ .

$m \angle BCD = 124^{\circ }$

$m \angle ADB = ?$

Answer

The rhombus referenced is below:

As a rhombus is a parallelogram, consecutive angles $\angle BCD$ and $\angle ADC$ are supplementary - that is,

$m \angle ADC + m \angle BCD = 180^{\circ }$ .

Set $m \angle BCD = 124^{\circ }$ and solve:

$m \angle ADC + 124^{\circ } = 180^{\circ }$

$m \angle ADC + 124^{\circ }- 124 ^{\circ } = 180^{\circ } - 124 ^{\circ }$

$m \angle ADC = 56 ^{\circ }$

A diagonal of a rhombus bisects the angles at its endpoints, so, specifically, $\overline{ BD}$ bisects $\angle ADC$ . Therefore,

$m \angle ADB = \frac{1}{2} m \angle ADC = \frac{1}{2} \cdot 56 ^{\circ } = 28^{\circ }$ .

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Question

Given: Rhombuses and .

$m \angle A = 60 ^{\circ }$ and $m \angle F = 120 ^{\circ }$

True, false, or undetermined: Rhombus $ABCD \sim$ Rhombus .

Answer

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 $s_{1}$ be the common sidelength of Rhombus and $s_{2}$ 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, $\frac{s_{1}}{s_{2}}$ .

Also, a rhombus being a parallelogram, its opposite angles are congruent, and its consecutive angles are supplementary. Therefore, since $m \angle A = 60 ^{\circ }$ , it follows that $m\angle C = 60 ^{\circ }$ , and $m \angle B = m \angle D = 180 ^{\circ } - 60^{\circ } = 120 ^{\circ }$ . By a similar argument, $m \angle H = m \angle F = 120 ^{\circ }$ and $m \angle E = m \angle G= 180 ^{\circ } - 120 ^{\circ } = 60^{\circ }$ . Therefore,

$\angle A \cong \angle E$

$\angle B \cong \angle F$

$\angle C \cong \angle F$

$\angle D \cong \angle H$

Since all corresponding sides are proportional and all corresponding angles are congruent, it holds that Rhombus $ABCD \sim$ Rhombus .

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Question

Given: Rhombuses and .

True, false, or undetermined: Rhombus $ABCD \cong$ Rhombus .

Answer

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 $s_{1}$ be the common sidelength of Rhombus and $s_{2}$ be the common sidelength of Rhombus , then, since , it follows that $s_{1} = s_{2}$ , 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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Question

Given that a rhombus has a perimeter of , find the length of one side of the rhombus.

Answer

The perimeter of a rhombus is equal to , where the length of one side of the rhombus.

Since , we can set up the following equation and solve for .

$124=4\times (S)$

$S=\frac{124}{4}=31$

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Question

A rhombus has an area of square units and a height of . Find the length of one side of the rhombus.

Answer

To find the length of a side of the rhombus, work backwards using the formula:

$A=(base\times height)$

Since we are given the area and the height we plug these values in and solve for the base.

$360=(base\times 8)$

$base=\frac{360}{8}=45$

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Question

A rhombus has an area of square units, and a height of . Find the length of one side of the rhombus.

Answer

To find the length of a side of the rhombus, work backwards using the area formula:

$Area=(base\times height)$

Since we are given the area and the height we plug these values in and solve for the base.

$69=(base\times 6)$

$base=\frac{69}{6}=11.5$

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Question

A rhombus has a perimeter of . Find the length of one side of the rhombus.

Answer

To solve for the length of one side of the rhombus, apply the perimeter formula:

,

the length of one side of the rhombus.

Since we are given the area we plug this value in and solve for .

$S=\frac{720}{4}=180$

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Question

A rhombus has an area of square units, and an altitude of . Find the length of one side of the rhombus.

Answer

Since the area is equal to square units, use the formula:

$A=(base\times altitude)$

Since we are given the area and the height, we plug these values in and solve for the base.

$39=(base\times 3)$

$base=\frac{39}{3}=13$

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Question

The perimeter of a rhombus is equal to . Find the length of one side of the rhombus.

Answer

Since

The solution is:

, where the length of one side of the rhombus.

Thus,

$S=\frac{520}{4}=130$

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Question

If a rhombus has a base that is $\frac{4}{3}$ times greater than the height and the height of the rhombus is equal to , find the length of one side of the rhombus.

Answer

To find the length of a side of the rhombus, multiply times $\frac{4}{3}$ .

Thus, the solution is:

$\frac{9}{1}\times\frac{4}{3}=\frac{36}{3}=12$

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Question

A rhombus has an area of square units, and an altitude of . What is the length of one side of the rhombus?

Answer

To find the length of a side of the rhombus, work backwards using the area formula:

$Area=(base\times height)$

Since we are given the area and the height, we plug these values into the equation and solve for the base.

$645=(base\times 15)$

$base=\frac{645}{15}=43$

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Question

Given that a rhombus has an area of square units and a height of , find the length of one side of the rhombus.

Answer

To find the length of a side of the rhombus, work backwards using the area formula:

$Area=(base\times height)$

Since we are given the area and the height, we plug these values into the equation and solve for the base.

$85=(base\times height)$

$85=(base\times 5)$

$base=\frac{85}{5}=17$

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