# Year 10+ Plane Geometry

### Chapters

### Quadrilaterals

# Quadrilaterals

`Quadrilaterals`

are plane (flat) shapes with 4 straight edges and 4 angles.

They are a type of polygon, so they need to be **closed** (the ends need to meet up).

Each quadrilateral has

- 4 straight edges
- 4 vertices (corners)
- An angle sum (sum of the interior angles) of \(360^\circ\).

The picture on the left shows a few different examples of quadrilaterals.

Some other names for quadrilaterals are `quadrangles`

and `tetragons`

.

## Some More Examples

The sum of the angles in each of these quadrilaterals is \(360^\circ\) (don't forget that a right angle (marked by the little square in the green quadrilateral) is \(90^\circ\)).

There are a number of different special classes of quadrilaterals. They include rectangles, squares, rhombuses, parallelograms, kites and trapeziums. Let's have a look at each of these special types individually, and then we'll look at how they all fit together.

### Rectangles

`Rectangles`

have

- 2 pairs of opposite sides that are equal and parallel (one pair is marked in green on the diagram, and the other is red).
- 4 right angles

### Squares

`Squares`

have

- 2 pairs of opposite sides that are parallel.
- All sides equal in length.
- 4 right angles

### Parallelograms

`Parallelograms`

have

- 2 pairs of opposite sides that are equal and parallel (one pair is marked in green on the diagram, and the other is red).
- 2 pairs of opposite angles that are equal (one pair is marked green, and the other is marked red)

### Rhombuses

`Rhombuses`

have

- 2 pairs of opposite sides that are parallel.
- 4 equal sides.
- 2 pairs of opposite angles that are equal (one pair is marked green, and the other is marked red)

**Note:** Rhombuses, squares and rectangles are all parallelograms. A **rectangle** is a **parallelogram**
with one angle that is a right angle. A **square** is a **parallelogram** with all sides equal, and one angle a right angle.
A **rhombus** is a **parallelogram** with all sides equal in length.

### Trapeziums and Trapezoids

A `trapezium`

is a quadrilateral that has

- One pair of opposite sides that is parallel.

`isosceles trapezium`

also has two pairs of equal angles as shown in the diagram:
A `trapezoid`

is any old quadrilateral that has no parallel sides.
**Fun Fact**: If you do a web search for trapezoid, you'll find the definition of a trapezium, because people in the US and Canada
call trapeziums trapezoids, and trapezoids trapeziums. The reason for this is due to an error in Hutton's Mathematical Dictionary, written in
1795, which was the first mathematical dictionary to appear in the US. Unfortunately, the error stuck, and now the definitions are reversed in North
America.

### Kites

`Kites`

have

- 2 pairs of adjacent sides that are equal (marked red and green on the diagram).
- Diagonals that intersect at right angles. One diagonal cuts the other one in half.
- One pair of equal and opposite angles.

### Irregular Quadrilaterals

Every quadrilateral except for a square is an irregular quadrilateral, so there are plenty of them. Even though a rhombus has four equal sides, not all of its angles are equal.

### Relationships Between Quadrilaterals

Squares are Rectangles, Rectangles and Squares are Parallelograms, Rhombuses and Parallelograms and so on. The diagram below shows you all the relationships between the different special quadrilaterals:

So, we can make the following statements:

- A parallelogram is a trapezium.
- A square is a rhombus.
- A rhombus is a kite.

### Quadrilaterals Are Polygons

Quadrilaterals are 4-sided polygons. See the article on polygons for more information.

Just like polygons, quadrilaterals can be `complex`

: this means that their sides can cross over like the ones in the diagram below.

### Description

In these chapters you will learn about plane geometry topics such as

- Area (Irregular polygons, plane shapes etc)
- Perimeter
- Conic sections (Circle, Ellipse, Hyperbola etc)
- Polygons (Congruent, polygons, similar, triangles etc)
- Transformations and symmetry (Reflection, symmetry, transformations etc)

etc

Even though these chapters are marked for Year 10 or higher students, several topics are for students in Year 8 or higher

### Audience

Year 10 or higher, suitable for Year 8 + students as well.

### Learning Objectives

Learn about Plane Geometry

Author: Subject Coach

Added on: 28th Sep 2018

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