Parallelogram to Rectangle Practice Problems and Solutions

Master proving parallelograms are rectangles with step-by-step practice exercises. Learn angle and diagonal theorems through interactive problems and solutions.

📚Master Converting Parallelograms to Rectangles with Interactive Practice
  • Prove parallelograms are rectangles using 90-degree angle theorem
  • Apply equal diagonal theorem to identify rectangular parallelograms
  • Calculate missing angles in parallelograms with one known right angle
  • Determine rectangle properties using opposite and adjacent angle relationships
  • Solve problems involving isosceles triangles formed by equal diagonals
  • Practice identifying rectangles from given parallelogram conditions

Understanding From a Parallelogram to a Rectangle

Complete explanation with examples

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.

Parallelogram to a rectangle

Detailed explanation

Practice From a Parallelogram to a Rectangle

Test your knowledge with 2 quizzes

AAABBBDDDCCC90°

The quadrilateral ABCD is a parallelogram.

\( ∢B=90° \)

Is it a rectangle?

Examples with solutions for From a Parallelogram to a Rectangle

Step-by-step solutions included
Exercise #1

Side DA is equal to side DE.

Is the parallelogram a rectangle?

454545E1E1E1E2E2E2AAABBBCCCDDDEEE

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.

Video Solution
Exercise #2

Is the parallelogram below a rectangle?

AAABBBCCCDDD3392

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

Video Solution
Exercise #3

Is the parallelogram below a rectangle?

AAABBBCCCDDD2664

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

Video Solution
Exercise #4

Determine whether the parallelogram below is a rectangle?

989898AAABBBCCCDDD

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

Video Solution
Exercise #5

AAABBBDDDCCC120°

Is this parallelogram a rectangle

Step-by-Step Solution

Answer:

No

Video Solution

Frequently Asked Questions

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

How do you prove a parallelogram is a rectangle?

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You can prove a parallelogram is a rectangle using two main theorems: 1) Show that one angle equals 90 degrees, or 2) Prove that the diagonals are equal in length. Since all parallelograms have opposite angles equal and adjacent angles supplementary, proving one right angle automatically makes all angles 90 degrees.

What is the difference between a parallelogram and rectangle?

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A rectangle is a special type of parallelogram where all angles are 90 degrees. While both shapes have opposite sides parallel and equal, rectangles have the additional property of right angles and equal diagonals, making them more restrictive than general parallelograms.

Why are diagonals equal in rectangles but not parallelograms?

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In rectangles, the right angles create congruent right triangles when diagonals are drawn, making the diagonals equal. In general parallelograms, the angles are not necessarily 90 degrees, so the triangles formed by diagonals are not congruent, resulting in unequal diagonal lengths.

What happens to angles when one angle in a parallelogram is 90 degrees?

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When one angle in a parallelogram equals 90 degrees, all angles become 90 degrees. This occurs because: 1) Opposite angles are equal, so the opposite angle is also 90°, 2) Adjacent angles are supplementary (sum to 180°), making the other two angles 90° each.

How do you find missing angles in a parallelogram that might be a rectangle?

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Use these parallelogram properties: opposite angles are equal, adjacent angles sum to 180°, and all four angles sum to 360°. If given one angle is 90°, then opposite angle = 90°, and adjacent angles = 180° - 90° = 90°. This confirms it's a rectangle.

What are the key properties to check when converting parallelogram to rectangle?

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Check these properties: 1) All angles equal 90 degrees, 2) Diagonals are equal in length, 3) Diagonals bisect each other at right angles, 4) Opposite sides remain parallel and equal. Any parallelogram meeting these conditions is automatically classified as a rectangle.

Can a parallelogram have equal diagonals without being a rectangle?

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No, if a parallelogram has equal diagonals, it must be a rectangle. This is one of the fundamental theorems: equal diagonals in a parallelogram force all angles to become 90 degrees, which is the defining characteristic of rectangles.

How do isosceles triangles form when proving parallelogram to rectangle using diagonals?

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When diagonals are equal in a parallelogram, they bisect each other creating four equal segments from the center. These equal segments form isosceles triangles with equal base angles. The symmetry of these triangles forces all parallelogram angles to equal 90°, proving it's a rectangle.

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