Parallelogram to Rhombus Practice Problems and Examples

Master identifying rhombus properties with step-by-step practice problems. Learn three key criteria: adjacent equal sides, perpendicular diagonals, and angle bisectors.

📚Master Parallelogram to Rhombus Identification
  • Identify when adjacent sides in a parallelogram make it a rhombus
  • Apply perpendicular diagonal criterion to prove rhombus properties
  • Use angle bisector diagonal method for rhombus identification
  • Solve congruent triangle problems using SAS criterion
  • Practice transitive relation with equal sides in quadrilaterals
  • Master three key criteria for parallelogram to rhombus conversion

Understanding From a Parallelogram to a Rhombus

Complete explanation with examples

You will be able to determine that the parallelogram is a rhombus if at least one of the following conditions is met:

  1. If in the parallelogram there is a pair of adjacent equal sides - it is a rhombus.
  2. If in the parallelogram the diagonals bisect each other, forming angles of 90o 90^o degrees, that is, they are perpendicular - it is a rhombus.
  3. If in the parallelogram one of the diagonals is the bisector - it is a rhombus.

Detailed explanation

Practice From a Parallelogram to a Rhombus

Test your knowledge with 2 quizzes

Look at the parallelogram below:

AAABBBDDDCCC

The diagonals form 2 pairs of different angles at the center of the parallelogram.

Is the parallelogram a rhombus?

Examples with solutions for From a Parallelogram to a Rhombus

Step-by-step solutions included
Exercise #1

AAABBBDDDCCC7575

Can the given parallelogram be considered a rhombus?

Step-by-Step Solution

The definition of a rhombus is "a parallelogram with equal sides"

In the parallelogram shown in the drawing, the adjacent sides are clearly not equal in length,

Therefore the parallelogram shown in the drawing cannot be considered a rhombus.

Therefore the correct answer is answer B.

Answer:

No

Video Solution
Exercise #2

Look at the parallelogram below:

AAABBBDDDCCC

If the diagonals cross at 90 degree angles at the center of the parallelogram.

Is this parallelogram considered a rhombus?

Step-by-Step Solution

The parallelogram whose diagonals are perpendicular to each other (meaning the angle between them is 90° 90\degree ) is a rhombus, therefore the given parallelogram is a rhombus.

Therefore, the correct answer is answer A.

Answer:

Yes.

Video Solution
Exercise #3

AAABBBDDDCCCCan the above parallelogram be considered a rhombus?

Step-by-Step Solution

The definition of a rhombus is "a quadrilateral with all equal sides"

Therefore, the square in the diagram is indeed a rhombus

Thus, the correct answer is answer A.

Answer:

True

Video Solution
Exercise #4

Given the parallelogram:

AAABBBDDDCCC149149

Is this parallelogram a rhombus?

Step-by-Step Solution

Answer:

Not true

Video Solution
Exercise #5

Given the parallelogram:

AAABBBDDDCCC9999

Is this parallelogram a rhombus?

Step-by-Step Solution

Answer:

True

Video Solution

Frequently Asked Questions

Everything you need to know about From a Parallelogram to a Rhombus

What are the three ways to prove a parallelogram is a rhombus?

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A parallelogram is a rhombus if: (1) it has a pair of adjacent equal sides, (2) its diagonals are perpendicular and bisect each other at 90°, or (3) one of its diagonals acts as an angle bisector. These three criteria are sufficient to identify any rhombus.

How do you prove adjacent sides make a parallelogram a rhombus?

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If two adjacent sides are equal (like AB = BC), then using the parallelogram property that opposite sides are equal, you can show all four sides are equal through transitive relation: AB = DC and AD = BC, therefore AB = DC = BC = AD.

Why do perpendicular diagonals make a parallelogram a rhombus?

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When diagonals are perpendicular in a parallelogram, you can prove adjacent sides are equal using triangle congruence (SAS). The perpendicular diagonals create congruent triangles, making corresponding sides equal, which satisfies the rhombus definition.

What does it mean for a diagonal to be an angle bisector in a rhombus?

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When a diagonal bisects an angle, it divides that angle into two equal parts. Using alternate angles between parallel sides and transitive relation, this creates equal angles that make opposite sides equal, proving the parallelogram is a rhombus.

Can you use just one criterion to prove a parallelogram is a rhombus?

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Yes, you only need to prove ONE of the three criteria: adjacent equal sides, perpendicular diagonals, or diagonal as angle bisector. Each criterion alone is sufficient to prove that a parallelogram is a rhombus.

What is the difference between a parallelogram and a rhombus?

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A rhombus is a special type of parallelogram where all four sides are equal. Every rhombus is a parallelogram, but not every parallelogram is a rhombus. The key difference is that rhombi have equal sides while general parallelograms only need opposite sides to be equal.

How do you use SAS congruence to prove perpendicular diagonals make a rhombus?

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With perpendicular diagonals, triangles ABE and BCE are congruent by SAS: AE = CE (diagonals bisect), angle AEB = angle BEC (both 90°), and BE is common. This proves AB = BC, making adjacent sides equal.

What memory trick helps remember rhombus identification criteria?

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Remember the three key terms: SIDES (adjacent equal sides), DIAGONALS (perpendicular diagonals), and ANGLES (diagonal as angle bisector). These correspond to the three main criteria for proving a parallelogram is a rhombus.

Continue Your Math Journey

Master From a Parallelogram to a Rhombus first, then advance to these related topics that build on your fraction skills

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