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:

Equilateral Triangle
Equilateral Triangle

Equilateral Triangle

Three equal sides

Three equal angles, always 60°
Isosceles Triangle
Isosceles Triangle

Isosceles Triangle

Two equal sides

Two equal angles
Scalene Triangle
Scalene Triangle

Scalene Triangle

No equal sides

No equal angles

What Type of Angle?

As the Angle Increases, the Name Changes:


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
less than 180°
Straight Angle
an angle that is 180° exactly
Reflex Angle
an angle that is greater than 180°


types of angle
types of angle

Combining the Names

Sometimes a triangle will have two names, for example:

Right Isosceles Triangle
Right Isosceles Triangle

Right Isosceles Triangle

Has a right angle (90°), and also two equal angles


Can you guess what the equal angles are?


You might also like to play with the Interactive Triangle.


Try working out angles in a triangle in this worksheet

or harder

Quadrilaterals

Quadrilaterals
Quadrilaterals

Quadrilateral just means "four sides"

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

Try for Yourself


(You can also play with Interactive Quadrilaterals)

Properties

  • Four sides (edges)
  • Four vertices (corners)
  • The interior angles add up to 360 degrees:
Quadrilateral Angles
Quadrilateral Angles
Try drawing a quadrilateral, and measure the angles. They should add to 360°


Try to work out the unknown angles of the quadrilaterals in this worksheet:

Types of Quadrilaterals


There are special types of quadrilateral:
Types of Quadrilateral
Types of Quadrilateral

Some types are also included in the definition of other types! For example a square, rhombus andrectangle are also parallelograms. See below for more details.
Let us look at each type in turn:

The Rectangle

Rectangle
Rectangle



external image quadrilateral-right-key.gif
means "right angle"
external image quadrilateral-equal-key.gifandexternal image quadrilateral-equal-key2.gif
show equal sides


A rectangle is a four-sided shape where every angle is a right angle (90°).
Also opposite sides are parallel and of equal length.

The Rhombus

Rhombus
Rhombus
A rhombus is a four-sided shape where all sides have equal length.
Also opposite sides are parallel and opposite angles are equal.
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.
A rhombus is sometimes called a diamond.

The Square

Square
Square



external image quadrilateral-right-key.gif
means "right angle"
external image quadrilateral-equal-key.gif
show equal sides


A square has equal sides and every angle is a right angle (90°)
Also opposite sides are parallel.
A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

The Parallelogram

Parallelogram
Parallelogram
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!

Example:


square
square

A parallelogram with:
  • all sides equal and
  • angles "a" and "b" as right angles
is a square!

The Trapezoid (UK: Trapezium)

Trapezoid (or Trapezium)
Trapezoid (or Trapezium)

Trapezoid
Isosceles Trapezoid
A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.
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.
And a trapezium (UK: trapezoid) is a quadrilateral with NO parallel sides:

Trapezoid
Trapezium
US:
a pair of parallel sides
NO parallel sides
UK:
NO parallel sides
a pair of parallel sides

The Kite

The Kite
The Kite
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.

Irregular Quadrilaterals

The only regular quadrilateral is a square. So all other quadrilaterals are irregular.

The "Family Tree" Chart

Quadrilateral definitions are inclusive.

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.")


This may seem odd, as in daily life we think of a square as not being a rectangle ... but in mathematics it is.
Using the chart below you can answer such questions as:
  • Is a Square a type of Rectangle? (Yes)
  • Is a Rectangle a type of Kite? (No)
Quadrilateral Classification
Quadrilateral Classification

Complex Quadrilaterals

Oh Yes! when two sides cross over, you call it a "Complex" or "Self-Intersecting" quadrilateral like these:
Complex Quadrilaterals
Complex Quadrilaterals
They still have 4 sides, but two sides cross over.

Polygon

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.

Play with Them

Now that you know the different types, you can play with the Interactive Quadrilaterals.

Other Names

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

regular_polygons.jpg
regular_polygons.jpg


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


3D Shapes

solids.gif
solids.gif



Here are some notes about adding angles in a triangle
external image pdf.png Adding angles in a triangle.pdf

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.
external image angle180.gif

30° + 150° = 180°

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

external image angle45.gif
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" ?

==|| external image angleb.gif
Angle b is simply 180° less the sum of the other angles.
Sum of known angles = 45° + 39° + 24° Sum of known angles = 108°
Angle b

180° − 108°Angle b

72°

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



Compass: North, South, East and West
external image compass-rose.gif

Directions on the "Compass Rose"

Compass Bearings

A Compass Bearing tells us
Direction
The 4 main directions are North, South, East and West(going clockwise they are NESW).

How to remember?
With North pointing ahead, "West" and "East" make the word "WE"

external image direction-nsew-.gif

Play The Game

Try the Direction Game.

In Between

Halfway between North and East is North-East (NE).
There is also South-East (SE), South-West (SW) and North-West (NW).
And in between all of those are:
  • NNE (North-North-East),
  • ENE (East-North-East),
  • ESE (East-South-East),
  • SSE (South-South-East),
  • SSW (South-South-West),
  • WSW (West-South-West),
  • WNW (West-North-West),
  • NNW (North-North-west)

Example: in the morning there was a strong North wind, but later it swung around to the North-East

Example: they were sailing mainly South-West, but sometimes a little towards South-South-West.

external image compass-bearing.gif

Three-Figure Bearings

Three-figure bearings are an alternative to compass bearings that are much more precise. They are measured in a special way:
  • Start measuring from the direction North
  • Measure clockwise
  • Give the bearing using three figures (or more than three if there's a decimal)
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.

Examples

The four main compass bearings (North, East, South and West) are multiples of 90°:
walk10
walk10

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:

walk12
walk12

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.

Example: South-West is 225 (in other words 225° clockwise from North)
Activity: A Walk in the Desert
2Measurement Index


Check out this site for notes and examples on bearings

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

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.

tessellation-1.jpg
tessellation-1.jpg
tessellation- shapes.png
tessellation- shapes.png
tessellation- fish.jpg
tessellation- fish.jpg