Intermediate Geometry : How to find if rhombuses are similar

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

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

Given: Rhombuses  and .

True, false, or undetermined: Rhombus  Rhombus .

Possible Answers:

True

False

Undetermined

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  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 .

Possible Answers:

Undetermined

True

False

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 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 .

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