From a Parallelogram to a Rectangle - Examples, Exercises and Solutions

Question Types:
From a Parallelogram to a Rectangle: Identifying and defining elements

From the Parallelogram to the Rectangle

Do you want to know how to prove that the parallelogram in front of you is actually a rectangle?
First, you should know that the formal definition of a rectangle is a parallelogram whose angle is 90o 90^o degrees.
Additionally, if the diagonals in parallelograms are equal, it is a rectangle.

That is, if you are given a parallelogram, you can prove it is a rectangle using one of the following theorems:

  • If a parallelogram has an angle of 90o 90^o degrees, it is a rectangle.
  • If the diagonals are equal in a parallelogram, it is a rectangle

We briefly remind you of the conditions for a parallelogram check:

  1. If in a quadrilateral where each pair of opposite sides are also parallel to each other, the quadrilateral is a parallelogram.
  2. If in a quadrilateral where each pair of opposite sides are also equal to each other, the quadrilateral is a parallelogram.
  3. If a quadrilateral has a pair of opposite sides that are equal and parallel, the quadrilateral is a parallelogram.
  4. If in a quadrilateral the diagonals cross each other, the quadrilateral is a parallelogram.
  5. If a quadrilateral has two pairs of equal opposite angles, the quadrilateral is a parallelogram.
Detailed explanation

Suggested Topics to Practice in Advance

  1. Rectangle
  2. From a Quadrilateral to a Rectangle

Practice From a Parallelogram to a Rectangle

Question 1

Is the parallelogram below a rectangle?

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Question 2

Is the parallelogram below a rectangle?

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Question 3

Is the parallelogram below a rectangle?

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Question 4

Side DA is equal to side DE.

Is the parallelogram a rectangle?

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Question 5

Perhaps all parallelogram is also a rectangle?

Examples with solutions for From a Parallelogram to a Rectangle

Exercise #1

Is the parallelogram below a rectangle?

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Video Solution

Step-by-Step Solution

Let's calculate angle B:

64+26=90 64+26=90

The parallelogram in the drawing is indeed a rectangle since a parallelogram with at least one right angle is a rectangle.

Answer

Yes

Exercise #2

Is the parallelogram below a rectangle?

AAABBBCCCDDD3392

Video Solution

Step-by-Step Solution

Let's calculate angle A:

92+33=125 92+33=125

The parallelogram in the diagram is not a rectangle since angle A is greater than 90 degrees, and in a rectangle all angles are right angles.

Answer

No

Exercise #3

Is the parallelogram below a rectangle?

989898AAABBBCCCDDD

Video Solution

Step-by-Step Solution

The parallelogram in the drawing cannot be a rectangle because in a rectangle all angles are right angles, meaning they are equal to 90 degrees.

Angle A is greater than 90 degrees.

Answer

No

Exercise #4

Side DA is equal to side DE.

Is the parallelogram a rectangle?

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Video Solution

Step-by-Step Solution

Looking at triangle DAE, we are given that DA equals DE, therefore the triangle is isosceles.

As a result, angles DAE and DEA are both equal to 45 degrees.

Now let's calculate angle D:

We know that the sum of angles in a triangle is 180 degrees, and since we have two angles of 45 degrees:

D+45+45=180 D+45+45=180

D+90=180 D+90=180

D=180−90 D=180-90

D=90 D=90

Since angle D is a right angle, the parallelogram is indeed a rectangle, according to the rule that if one angle in a parallelogram is a right angle, the parallelogram is a rectangle.

Answer

Yes.

Exercise #5

Perhaps all parallelogram is also a rectangle?

Video Solution

Answer

No, a rectangle necessarily has angles of 90°.

Question 1

AAABBBDDDCCC45°45°

Is this parallelogram a rectangle?

Question 2

AAABBBDDDCCC90°

The quadrilateral ABCD is a parallelogram.

\( ∢B=90° \)

Is it a rectangle?

Question 3

AAABBBDDDCCC

The quadrilateral ABCD is a parallelogram.

AD = CB

Is the parallelogram a rectangle?

Question 4

AAABBBDDDCCC120°

Is this parallelogram a rectangle

Question 5

Given the following parallelogram:

AAABBBCCCDDD30°70°

Is it a rectangle?

Exercise #6

AAABBBDDDCCC45°45°

Is this parallelogram a rectangle?

Video Solution

Answer

Yes

Exercise #7

AAABBBDDDCCC90°

The quadrilateral ABCD is a parallelogram.

∢B=90° ∢B=90°

Is it a rectangle?

Video Solution

Answer

Yes.

Exercise #8

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The quadrilateral ABCD is a parallelogram.

AD = CB

Is the parallelogram a rectangle?

Video Solution

Answer

Yes, a parallelogram with equal diagonals is a rectangle.

Exercise #9

AAABBBDDDCCC120°

Is this parallelogram a rectangle

Video Solution

Answer

No

Exercise #10

Given the following parallelogram:

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Is it a rectangle?

Video Solution

Answer

No