Parallelogram Analysis: Determining if a 45° Quadrilateral is a Rectangle

Q: Why are the base angles equal in triangle ABD?

In any isosceles triangle, the base angles (angles opposite the equal sides) are always equal. Since $ AD = AB $, angles ABD and ADB must both equal 45°.

Q: What if I can't see the 45° angle clearly in the diagram?

The angle is given in the problem as 45°. Trust the given information and use it in your calculations, even if the diagram isn't perfectly to scale.

Q: Do I need to prove all angles are 90° to show it's a rectangle?

No! In a parallelogram, if you can prove just one angle is 90°, then all four angles must be 90°. This is because opposite angles are equal and consecutive angles are supplementary.

Q: Could this parallelogram be a square instead of just a rectangle?

A square is a special type of rectangle where all sides are equal. From the given information, we can only confirm it's a rectangle. To prove it's a square, we'd need additional information about side lengths being equal.

Rectangle Identification with Isosceles Triangle Analysis

A parallelogram is shown below.

Determine whether the following parallelogram a rectangle:

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A parallelogram is shown below.

Determine whether the following parallelogram a rectangle:

Step-by-step solution

According to the data shown in the drawing, we know that $AD=AB$

$BC=CD$

This means that triangles BCD and DAB are isosceles triangles.

Since we are given that angle ABD equals 45 degrees, angle ADB also equals 45 degrees (in an isosceles triangle, the base angles are equal)

We can calculate angle A since in a triangle the sum of angles equals 180:

$A+45+45=180$

$A+90=180$

$A=180-90$

$A=90$

Given that angle A is a right angle in the parallelogram, we can determine that the parallelogram is a rectangle according to the rule:

A parallelogram with at least one right angle is a rectangle.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Rule: A parallelogram with one right angle is a rectangle
Technique: Use isosceles triangle base angles: if $\angle ABD = 45°$ , then $\angle ADB = 45°$
Check: Verify angle sum in triangle: $45° + 45° + 90° = 180°$ ✓

Common Mistakes

Avoid these frequent errors

Assuming all angles must be 90° to prove it's a rectangle
Don't try to measure or calculate all four angles = wasted time and confusion! You only need to prove ONE angle is 90° in a parallelogram. Always use the rule: one right angle in a parallelogram makes it a rectangle.

Practice Quiz

Test your knowledge with interactive questions

True or false?

One of the angles in a rectangle may be an acute angle.

FAQ

Everything you need to know about this question

How do I know this parallelogram has isosceles triangles?

The diagram shows diagonal BD creating triangles. Since opposite sides in a parallelogram are equal, we have $AD = BC$ and $AB = CD$ , making triangles ABD and CBD isosceles.

Why are the base angles equal in triangle ABD?

In any isosceles triangle, the base angles (angles opposite the equal sides) are always equal. Since $AD = AB$ , angles ABD and ADB must both equal 45°.

What if I can't see the 45° angle clearly in the diagram?

The angle is given in the problem as 45°. Trust the given information and use it in your calculations, even if the diagram isn't perfectly to scale.

Do I need to prove all angles are 90° to show it's a rectangle?

No! In a parallelogram, if you can prove just one angle is 90°, then all four angles must be 90°. This is because opposite angles are equal and consecutive angles are supplementary.

Could this parallelogram be a square instead of just a rectangle?

A square is a special type of rectangle where all sides are equal. From the given information, we can only confirm it's a rectangle. To prove it's a square, we'd need additional information about side lengths being equal.

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