Geometry

=For this topic you will need to know:=
 * How to name, draw and measure angles
 * How to classify polygons, triangles, quadrilaters
 * Angle rules: straight line, angles in a triangle, angles in quadrilaterals, vertically opposite and at a point
 * Bearing and compass points

Equilateral, Isosceles and Scalene Triangles:
There are three special names given to triangles that tell how many sides (or angles) are equal.

There can be **3**, **2** or **no** equal sides/angles: 
 * [[image:http://www.mathsisfun.com/images/triangle-equilateral.gif width="150" height="106" caption="Equilateral Triangle"]] || === Equilateral Triangle ===
 * Three** equal sides


 * Three** equal angles, always 60° ||
 * [[image:http://www.mathsisfun.com/images/triangle-isosceles.gif width="149" height="127" caption="Isosceles Triangle"]] || === Isosceles Triangle ===
 * Two** equal sides


 * Two** equal angles ||
 * [[image:http://www.mathsisfun.com/images/triangle-scalene.gif width="150" height="63" caption="Scalene Triangle"]] || === Scalene Triangle ===
 * No** equal sides


 * No** equal angles ||

As the Angle Increases, the Name Changes:
 less than 180° ||^  ||
 * ~ Type of Angle ||~ Description ||  ||
 * [|Acute Angle] || an angle that is less than 90° ||^  ||
 * [|Right Angle] || an angle that is 90° exactly ||^  ||
 * [|Obtuse Angle] || an angle that is greater than 90° but
 * [|Straight Angle] || an angle that is 180° exactly ||^  ||
 * [|Reflex Angle] || an angle that is greater than 180° ||^  ||



Combining the Names
Sometimes a triangle will have two names, for example:  Has a right angle (90°), and also two equal angles
 * [[image:http://www.mathsisfun.com/images/triangle-right-isosceles.gif width="150" height="113" caption="Right Isosceles Triangle"]] || === Right Isosceles Triangle ===

Can you guess what the equal angles are? ||  ==You might also like to play with the [|Interactive Triangle]. ==

=Quadrilaterals =
 * [[image:http://www.mathsisfun.com/images/quadrilaterals.gif width="345" height="208" caption="Quadrilaterals"]] || Quadrilateral just means "four sides"

(//quad// means four, //lateral// means side). But the sides have to be **straight**, and it has to be **2-dimensional**. ||
 * Any four-sided shape is a Quadrilateral**.

Try for Yourself
(You can also play with [|Interactive Quadrilaterals])

Properties
Try drawing a quadrilateral, and measure the angles. They should add to **360°**
 * Four sides (edges)
 * Four vertices (corners)
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">The interior angles add up to **360 degrees**:

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Types of Quadrilaterals
<span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> There are special types of quadrilateral: <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Some types are also included in the definition of other types! For example a **square**, **rhombus** and**rectangle** are also **//parallelograms//**. [|See below] for more details. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Let us look at each type in turn:

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">The Rectangle
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A [|rectangle] is a four-sided shape where every angle is a [|right angle] (90°). <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Also **opposite sides** are [|parallel] and of equal length.
 * [[image:http://www.mathsisfun.com/images/quadrilateral-rectangle.gif width="193" height="132" caption="Rectangle"]] ||  ||   ||
 * ^  || [[image:http://www.mathsisfun.com/images/quadrilateral-right-key.gif width="19" height="22"]] || //means "right angle"// ||
 * ^  || [[image:http://www.mathsisfun.com/images/quadrilateral-equal-key.gif width="11" height="19"]]and[[image:http://www.mathsisfun.com/images/quadrilateral-equal-key2.gif width="19" height="19"]] || //show equal sides// ||

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">The Rhombus
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A [|rhombus] is a four-sided shape where all sides have equal length. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Also opposite sides are parallel //and// opposite angles are equal. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Another interesting thing is that the diagonals (dashed lines in second figure) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A rhombus is sometimes called a diamond.

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">The Square
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A [|square] has equal sides and every angle is a right angle (90°) <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Also opposite sides are parallel. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A square also fits the definition of a **rectangle** (all angles are 90°), and a **rhombus** (all sides are equal length).
 * [[image:http://www.mathsisfun.com/images/quadrilateral-square.gif width="138" height="133" caption="Square"]] ||  ||   ||
 * ^  || [[image:http://www.mathsisfun.com/images/quadrilateral-right-key.gif width="19" height="22"]] || //means "right angle"// ||
 * ^  || [[image:http://www.mathsisfun.com/images/quadrilateral-equal-key.gif width="19" height="19"]] || //show equal sides// ||

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">The Parallelogram
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A [|parallelogram] has opposite sides parallel and equal in length. Also opposite angles are equal (angles "a" are the same, and angles "b" are the same).

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms! <span style="background-color: #d8d8ff; display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">

Example:
is a **square**! ||
 * [[image:http://www.mathsisfun.com/geometry/images/regular-quadrilateral-sm.gif width="40" height="40" caption="square"]] || A **parallelogram** with:
 * all sides equal and
 * angles "a" and "b" as right angles

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">The Trapezoid (UK: Trapezium)
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A [|trapezoid] (called a trapezium in the UK) has a pair of opposite sides parallel. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">It is called an **Isosceles** trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">And a **trapezium** (UK: trapezoid) is a quadrilateral with NO parallel sides:
 * [[image:http://www.mathsisfun.com/images/quadrilateral-trapezium.gif width="394" height="87" caption="Trapezoid (or Trapezium)"]] ||
 * Trapezoid ||  Isosceles Trapezoid  ||
 * || Trapezoid || Trapezium ||
 * US: || a pair of parallel sides || NO parallel sides ||
 * UK: || NO parallel sides || a pair of parallel sides ||

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">The Kite
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Hey, it looks like a [|kite]. It has two pairs of sides. Each pair is made up of adjacent sides that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other. ... and that's it for the special quadrilaterals.

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Irregular Quadrilaterals
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">The only [|regular] quadrilateral is a square. So all other quadrilaterals are **irregular**.

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">The "Family Tree" Chart
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Quadrilateral definitions are **inclusive**. <span style="background-color: #d8d8ff; display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">

Example: a square is also a rectangle.
So we **include** a square in the definition of a rectangle. //(We **don't** say "Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.")// <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">This may seem odd, as in daily life we think of a square as **not** being a rectangle ... but in mathematics it **is**. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Using the chart below you can answer such questions as:
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Is a Square a type of Rectangle? (Yes)
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Is a Rectangle a type of Kite? (No)

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Complex Quadrilaterals
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Oh Yes! when two sides cross over, you call it a "Complex" or "Self-Intersecting" quadrilateral like these: They still have 4 sides, but two sides cross over.

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Polygon
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A quadrilateral is a [|polygon]. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on.

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Play with Them
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Now that you know the different types, you can play with the [|Interactive Quadrilaterals].

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Other Names
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">A quadrilateral can sometimes be called: = =
 * a **Quadrangle** ("//four angles//"), so it sounds like "triangle"
 * a **Tetragon** ("//four// and //polygon//"), so it sounds like "pentagon", "hexagon", etc.

= ** Polygons ** =

Irregular polygons are where not all the sides (and angles) are the same.

= 3D Shapes  =

Here are some notes about adding angles in a triangle = = =Angles On One Side of A Straight Line=

Angles on one side of a straight line will always add to **180 degrees**.
If a line is split into 2 and you know one angle you can always find the other one.

30° + 150° = **180°**

Example: If we know one angle is 45° what is angle **"a"** ?

 * [[image:http://www.mathsisfun.com/images/angle45.gif width="188" height="110"]] || Angle **a** is 180° − 45° = **135°** ||

This method can be used for several angles on one side of a straight line.

Example: What is angle **"b"** ?
==|| Sum of known angles = 45° + 39° + 24° Sum of known angles = 108° Angle **b** =180° − 108°Angle **b**= 72° ||
 * Angle **b** is simply 180° less the sum of the other angles.


 * Click on this[| link]for more examples and games**
 * Or this [|link] to check out 'Angles at a Point'**

<span style="color: #003399; display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 23px; text-align: center;">Compass: North, South, East and West <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">

<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Directions on the "Compass Rose"

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Compass Bearings
<span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 17px;">A Compass Bearing tells us Direction <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">The 4 main directions are **North**, **South**, **East** and **West**(going clockwise they are NESW). <span style="background-color: #fff8dc; display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> With North pointing ahead, "West" and "East" make the word "WE" Try the [|Direction Game]. ||
 * How to remember?**
 * [[image:http://www.mathsisfun.com/games/images/direction-nsew-.gif width="100" height="100" link="http://www.mathsisfun.com/games/direction-nsew-.html"]] || === Play The Game ===

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">In Between
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Halfway between North and East is North-East (NE). <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">There is also South-East (SE), South-West (SW) and North-West (NW). <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">And in between all of those are: <span style="background-color: #d8d8ff; display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> Example: in the morning there was a strong North wind, but later it swung around to the North-East <span style="background-color: #d8d8ff; display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> Example: they were sailing mainly South-West, but sometimes a little towards South-South-West. <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">NNE (North-North-East),
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">ENE (East-North-East),
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">ESE (East-South-East),
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">SSE (South-South-East),
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">SSW (South-South-West),
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">WSW (West-South-West),
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">WNW (West-North-West),
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">NNW (North-North-west)

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Three-Figure Bearings
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Three-figure bearings are an alternative to compass bearings that are much more precise. They are measured in a special way: <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Airline pilots and ships' helmsmen use three-figure bearings so that they can point their craft in exactly the right direction to safely reach their destination. <span style="background-color: #d8d8ff; display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Start measuring from the direction North
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Measure clockwise
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Give the bearing using three figures (or more than three if there's a decimal)

Examples
The four main compass bearings (North, East, South and West) are multiples of 90°: Notice that east, for example is 090° rather than 90° because it is given as three figures. The advantage of three-figure bearings is that they describe any direction uniquely: Note that the last one has four figures (three in front of the decimal point and one after) but it is still a "three-figure bearing", the .4 just gives more accuracy. <span style="background-color: #d8d8ff; display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> Example: South-West is 225 (in other words 225° clockwise from North) <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 17px;"> [|Activity: A Walk in the Desert] <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 17px;"> [|2][|Measurement Index] <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 17px;"> <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 17px;"> =Bearing lesson:= Try the questions on this [|link] See if you can help this person find the answer he is looking for - [|you will need to read this carefully] Now do the 'bearing' worksheet
 * Check out this [|site] for notes and examples on bearings**

=Transformation lesson:= Read about transformations [|here] Now read and do the lessons on [|Rotation] Read and do the lessons on [|Reflection] Read and do the lessons on [|Translation] Read and do the lessons on [|Resizing]

__ HOLIDAY TASK: Create a tessellation (due on your first day back from holiday) __
== Below are a couple of examples; they look like wall paper patterns, you can use polygons (shapes with straight sides) or other shapes. You can find more shapes by 'googling' Tessellations and selecting 'images'. This [|website] shows you how to create your own tessellation. So does this [|one]. You can find your own websites too :) == ==Here is a [|youtube video] on how to make a tessellation.==