# How to Understand the Fundamental Properties of Parallelograms

Everyone has seen shapes with four sides, but parallelograms are a bit special. They stand out because their opposite sides are always parallel to each other. But that's not all that makes them interesting. In this guide, we're going to take a closer look at parallelograms, diving into their unique features and properties. We'll break down the rules that govern them and provide clear explanations to make them easy to understand. So, whether you're studying for a test, teaching a class, or just curious about shapes, this exploration into parallelograms will offer valuable insights. Ready to learn more?

## Step-by-step Guide: Properties of Parallelograms

**Definition**:

What exactly is a parallelogram? At its core, a parallelogram is a quadrilateral (a shape with four sides) where opposite sides are perfectly parallel.

**Opposite Sides are Buddies**:

- Not only are the opposite sides of a parallelogram parallel, but they’re also of equal length, making them congruent.

**Opposite Angles, Twin Angles**:

- Just like the opposite sides, the angles sitting opposite each other in a parallelogram are equal or congruent.

**Consecutive Angles Play Nice Together**:

- The consecutive (or neighboring) angles in a parallelogram are supplementary. Put simply, if you add their measures, you’ll get \(180^\circ\).

**Diagonals’ Special Traits**:

- The diagonals of a parallelogram have a knack for cutting each other exactly in half.
- However, unlike some of their quadrilateral cousins, the diagonals in a parallelogram aren’t always the same length.

**Calculating the Area**:

Want to find out how much space is inside a parallelogram? The formula is:

\(A = b \times h \)

Here, \(b\) stands for the length of the base, and \(h\) represents the perpendicular height (or altitude) from the opposite side to this base.

### Examples

**Example 1:** Discovering the Diagonals

In parallelogram \(ABCD\), if the diagonal \(AC\) spans \(10\) units and \(BD\) stretches over \(8\) units, how long are the segments \(AE\) and \(EC\) when \(E\) is the exact point where \(AC\) and \(BD\) cross paths?**Solution:**

Given the nature of parallelograms, their diagonals bisect each other. So,

\(AE = EC = \frac{AC}{2} = \frac{10 \text{ units}}{2} = 5\) units.

**Example 2: **Finding the Area**Question:** Given a parallelogram \(EFGH\) with its base \(EF\) measuring \(6\) units, and a height that runs \(4\) units perpendicular from \(G\) to \(EF\), can you compute the area?**Solution:**

Utilize the formula:

\(A = b \times h\)

\(A = 6 \text{ units} \times 4 \text{ units} = 24\) square units.

### Practice Questions:

- In a parallelogram, if one angle measures \(70^\circ\), can you figure out the measure of its opposite angle?
- With a side length of \(9\) units and a perpendicular height of \(7\) units from that side, how much space does a parallelogram occupy?
- Inside parallelogram \(LMNO\), if the diagonal \(LN\) stretches out to \(14\) units, can you determine the lengths of segments \(LP\) and \(PN\) when \(P\) marks the spot where the diagonals meet?

**Answers:**

- The opposite angles of a parallelogram are congruent. So, it’s \(70^\circ\).
- The area totals \(63\) square units.
- Since the diagonals in a parallelogram bisect each other, \(LP = PN = 7\) units.

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