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Intermediate Geometry : Quadrilaterals

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

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Example Question #1 : Rhombuses

Given: Rhombuses ABCD and EFGH .

AB = EF

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

Possible Answers:

False

Undetermined

True

Correct answer:

Undetermined

Explanation:

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 ABCD and $s_{2}$ be the common sidelength of Rhombus EFGH , then, since AB = EF , 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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Example Question #2 : How To Find If Rhombuses Are Similar

Given: Rhombuses ABCD and EFGH .

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

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

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 $s_{1}$ be the common sidelength of Rhombus ABCD and $s_{2}$ be the common sidelength of Rhombus EFGH , 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 EFGH .

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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 124 degrees and the other angle must be degrees.

Both of the remaining angles are 124 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 are 124 degrees.

Explanation:

The four interior angles in any rhombus must have a sum of 360 degrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum of 180 degrees.

Thus, if a rhombus has two interior angles of degrees, there must also be two angles that equal:

180-56=124

Check:

56+56+124+124=360

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Example Question #1 : How To Find An Angle In A Rhombus

Rhombus_missing_angle_dos

Using the rhombus above, find the sum of angles and

Possible Answers:

$79^\circ$

$158^\circ$

$202^\circ$

$101^\circ$

Correct answer:

$158^\circ$

Explanation:

The four interior angles in any rhombus must have a sum of 360 degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of 180 degrees.

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

$79+79=158^\circ$

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Example Question #3 : How To Find An Angle In A Rhombus

Rhombus_missing_angle_dos

In the above rhombus, angle C=79 degrees. Find the sum of angles and

Possible Answers:

$110^\circ$

$101^\circ$

$158^\circ$

$202^\circ$

Correct answer:

$202^\circ$

Explanation:

The four interior angles in any rhombus must have a sum of 360 degrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum of 180 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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Example Question #4 : How To Find An Angle In A Rhombus

Vt_vt_rhomb_tres

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

Possible Answers:

$107^\circ$

$126^\circ$

$63^\circ$

$117^\circ$

Correct answer:

$117^\circ$

Explanation:

The four interior angles in any rhombus must have a sum of 360 degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of 180 degrees.

Since angle is adjacent to angle , they must have a sum of 180 degrees.

The solution is:

$180-63=117^\circ$

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Example Question #5 : How To Find An Angle In A Rhombus

Vt_vt_rhomb_tres

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

Possible Answers:

$140^\circ$

$126^\circ$

$120^\circ$

$117^\circ$

Correct answer:

$126^\circ$

Explanation:

The four interior angles in any rhombus must have a sum of 360 degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of 180 degrees.

Angles and are opposite interior angles, so they must have equivalent measurements.

The sum is:

$63+63=126^\circ$

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Example Question #1 : Quadrilaterals

Vt_vt_rhomb_tres

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

Possible Answers:

$203^\circ$

$117^\circ$

$126^\circ$

$234^\circ$

Correct answer:

$234^\circ$

Explanation:

The four interior angles in any rhombus must have a sum of 360 degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of 180 degrees.

Since, both angles and are adjacent to angle --find the measurement of one of these two angles by: 180-63=117 .

Angle and angle must each equal 117 degrees. So the sum of angles and Y=117+117=234 degrees.

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Example Question #7 : How To Find An Angle In A Rhombus

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

True or false: Parallelogram ABCD 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 ABCD with diagonals $\overline{AC}$ and $\overline{BD}$ intersecting at point .

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

Possible Answers:

True

False

Correct answer:

True

Explanation:

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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Intermediate Geometry : Quadrilaterals

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : Rhombuses

Example Question #2 : How To Find If Rhombuses Are Similar

Example Question #1 : How To Find An Angle In A Rhombus

Example Question #1 : How To Find An Angle In A Rhombus

Example Question #3 : How To Find An Angle In A Rhombus

Example Question #4 : How To Find An Angle In A Rhombus

Example Question #5 : How To Find An Angle In A Rhombus

Example Question #1 : Quadrilaterals

Example Question #7 : How To Find An Angle In A Rhombus

Example Question #1 : How To Find An Angle In A Rhombus

All Intermediate Geometry Resources

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