Rectangle ABCD: Proving Diagonal Equality in Quadrilaterals

Q: Why are rectangle diagonals equal but not parallelogram diagonals?

A rectangle is a special parallelogram with four right angles. These right angles make the diagonal triangles congruent by SAS (Side-Angle-Side), ensuring equal diagonal lengths!

Q: Do all quadrilaterals have equal diagonals?

No! Only rectangles and squares have equal diagonals. Regular parallelograms, rhombi, and trapezoids generally have unequal diagonal lengths.

Q: How can I prove the diagonals are equal?

Use triangle congruence! Triangles $ ABC $ and $ BAD $ are congruent by SAS: two sides equal (rectangle sides) and included right angle.

Q: Is this property useful in real problems?

Absolutely! Diagonal equality helps solve for unknown side lengths, verify rectangle properties, and calculate areas using coordinate geometry.

Q: What about squares - do their diagonals work the same way?

Yes! A square is a special rectangle, so its diagonals are also equal. Plus, square diagonals are perpendicular and bisect each other at right angles.

Rectangle Properties with Diagonal Equality

True or false:

The diagonals of rectangle ABCD are equal.

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

True or false:

The diagonals of rectangle ABCD are equal.

Step-by-step solution

To solve this problem, we'll follow these steps:

Identify the diagonal lines of the rectangle $AC$ and $BD$ .
Apply the diagonal property theorem for rectangles.

Let's go through the solution:

A rectangle in Euclidean geometry has several properties, one of which is that its diagonals are equal in length. This can be derived from the fact that opposite sides of a rectangle are equal and parallel, making it a specific type of parallelogram. Thus, both diagonals split the rectangle into congruent triangles, ensuring they are of equal length.

For rectangle $ABCD$ , diagonals $AC$ and $BD$ must be equal due to this inherent geometric property.

Therefore, the statement that "the diagonals of rectangle ABCD are equal" is True.

Final Answer

True

Key Points to Remember

Essential concepts to master this topic

Rule: Rectangle diagonals are always equal in length
Technique: Use diagonal property: $AC = BD$ for rectangle ABCD
Check: Both diagonals create congruent right triangles with equal hypotenuses ✓

Common Mistakes

Avoid these frequent errors

Confusing rectangles with general parallelograms
Don't assume all parallelograms have equal diagonals = wrong generalization! Only rectangles (and squares) guarantee diagonal equality because of their right angles. Always identify the specific quadrilateral type first.

Practice Quiz

Test your knowledge with interactive questions

True or false:

The sum of the angles of a rectangle is 360.

FAQ

Everything you need to know about this question

Why are rectangle diagonals equal but not parallelogram diagonals?

A rectangle is a special parallelogram with four right angles. These right angles make the diagonal triangles congruent by SAS (Side-Angle-Side), ensuring equal diagonal lengths!

Do all quadrilaterals have equal diagonals?

No! Only rectangles and squares have equal diagonals. Regular parallelograms, rhombi, and trapezoids generally have unequal diagonal lengths.

How can I prove the diagonals are equal?

Use triangle congruence! Triangles $ABC$ and $BAD$ are congruent by SAS: two sides equal (rectangle sides) and included right angle.

Is this property useful in real problems?

Absolutely! Diagonal equality helps solve for unknown side lengths, verify rectangle properties, and calculate areas using coordinate geometry.

What about squares - do their diagonals work the same way?

Yes! A square is a special rectangle, so its diagonals are also equal. Plus, square diagonals are perpendicular and bisect each other at right angles.

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