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Intermediate Geometry : How to find if rhombuses are similar

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Example Question #1 : Rhombuses

Given: Rhombuses ABCD $\displaystyle ABCD$ and EFGH $\displaystyle EFGH$.

AB = EF $\displaystyle AB = EF$

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

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 $s_{1}$ $\displaystyle s_{1}$ be the common sidelength of Rhombus ABCD $\displaystyle ABCD$ and $s_{2}$ $\displaystyle s_{2}$ be the common sidelength of Rhombus EFGH $\displaystyle EFGH$, then, since AB = EF $\displaystyle AB = EF$, it follows that $s_{1} = s_{2}$ $\displaystyle s_{1} = s_{2}$, 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 #1 : How To Find If Rhombuses Are Similar

Given: Rhombuses ABCD $\displaystyle ABCD$ and EFGH $\displaystyle EFGH$.

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

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

Possible Answers:

False

Undetermined

True

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

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

$\angle A \cong \angle E$ $\displaystyle \angle A \cong \angle E$

$\angle B \cong \angle F$ $\displaystyle \angle B \cong \angle F$

$\angle C \cong \angle F$ $\displaystyle \angle C \cong \angle F$

$\angle D \cong \angle H$ $\displaystyle \angle D \cong \angle H$

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

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Intermediate Geometry : How to find if rhombuses are similar

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

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Example Question #1 : Rhombuses

Example Question #1 : How To Find If Rhombuses Are Similar

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