Parallelogram to Rectangle: When DA = DE in a Four-Sided Shape

Q: Why does having equal sides DA = DE help us?

When two sides of a triangle are equal, it creates an isosceles triangle. This means the angles opposite those equal sides must also be equal - both 45° in this case!

Q: How do I know the angles are 45° each?

Look at the diagram! The angles DAE and DEA are marked as 45°. In an isosceles triangle, the base angles are always equal to each other.

Q: Why does one 90° angle make the whole parallelogram a rectangle?

In a parallelogram, opposite angles are equal and adjacent angles are supplementary (add to 180°). If one angle is 90°, then all four angles must be 90°!

Q: Could this parallelogram NOT be a rectangle?

No! Once we prove angle D is 90°, the parallelogram must be a rectangle. There's no other possibility when we have these given conditions.

Parallelogram Properties with Isosceles Triangle Analysis

Side DA is equal to side DE.

Is the parallelogram a rectangle?

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Understand the problem

Side DA is equal to side DE.

Is the parallelogram a rectangle?

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:

$D+45+45=180$

$D+90=180$

$D=180-90$

$D=90$

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.

Final Answer

Yes.

Key Points to Remember

Essential concepts to master this topic

Rule: Isosceles triangle creates equal base angles (45° each)
Technique: Calculate third angle: $45 + 45 + D = 180$ , so $D = 90°$
Check: If one angle in parallelogram is 90°, then it's a rectangle ✓

Common Mistakes

Avoid these frequent errors

Assuming equal sides automatically make a rectangle
Don't think DA = DE means rectangle without checking angles = wrong conclusion! Equal sides only tell us about the triangle, not the parallelogram. Always calculate the angle first using triangle properties, then apply parallelogram rules.

Practice Quiz

Test your knowledge with interactive questions

The quadrilateral ABCD is a parallelogram.

$$ ∢B=90° $$

Is it a rectangle?

FAQ

Everything you need to know about this question

Why does having equal sides DA = DE help us?

When two sides of a triangle are equal, it creates an isosceles triangle. This means the angles opposite those equal sides must also be equal - both 45° in this case!

How do I know the angles are 45° each?

Look at the diagram! The angles DAE and DEA are marked as 45°. In an isosceles triangle, the base angles are always equal to each other.

Why does one 90° angle make the whole parallelogram a rectangle?

In a parallelogram, opposite angles are equal and adjacent angles are supplementary (add to 180°). If one angle is 90°, then all four angles must be 90°!

What if the triangle wasn't isosceles?

Then we couldn't determine if it's a rectangle! We need the equal sides to create equal base angles, which then lets us calculate the third angle as 90°.

Could this parallelogram NOT be a rectangle?

No! Once we prove angle D is 90°, the parallelogram must be a rectangle. There's no other possibility when we have these given conditions.

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