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Diagonals of Parallelograms

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Diagonals of Parallelograms

Study Guide

Key Definition

In a parallelogram, the diagonals bisect each other, meaning they divide each other into two equal parts. This can be expressed as $\overline{AE} \cong \overline{EC}$ and $\overline{BE} \cong \overline{ED}$.

Important Notes

The diagonals of a parallelogram intersect at their midpoints.
$\overline{AC}$ and $\overline{BD}$ are diagonals of the parallelogram.
If a quadrilateral's diagonals bisect each other, it is a parallelogram.
The intersection point of diagonals is the center of gravity of the parallelogram.
$\overline{AE} \cong \overline{EC}$ and $\overline{BE} \cong \overline{ED}$

Mathematical Notation

$\overline{AB}$ denotes a line segment between A and B

$\cong$ represents congruence

$\angle$ represents an angle

$\triangle$ denotes a triangle

$\frac{a}{b}$ represents a fraction

Remember to use proper notation when solving problems

Why It Works

Consider parallelogram ABCD with diagonals AC and BD intersecting at E. In triangles AEB and CED, AB = CD (opposite sides), EB = EB (common side), and ∠AEB = ∠CED (vertical angles). By SAS (Side-Angle-Side), these triangles are congruent, so $AE = EC$ and $BE = ED$, proving the diagonals bisect each other.

Remember

In a parallelogram, diagonals always bisect each other. This is a key property that can be used to identify and prove a shape is a parallelogram.

Quick Reference

Diagonal Bisect Property:$\overline{AE} \cong \overline{EC}$, $\overline{BE} \cong \overline{ED}$

Understanding Diagonals of Parallelograms

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In a parallelogram, the diagonals bisect each other, meaning they split into equal parts at the intersection point.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If the diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point E, what can we say about $\overline{AE}$ and $\overline{EC}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Consider a quadrilateral (e.g., a kite). If its diagonals bisect each other, is the shape necessarily a parallelogram? (Here the given condition of both diagonals bisecting overrides the usual kite property.)

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a quadrilateral with diagonals $\overline{AC}$ and $\overline{BD}$ that bisect each other. Prove that the quadrilateral is a parallelogram.

Challenge Quiz

Single Choice Quiz

Advanced

In a parallelogram, if one diagonal is twice as long as the other, what is the relationship between the parts created by the diagonals?

Recap

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Review key concepts and takeaways