… Protractor practice. Always start counting from zero so that you use the correct scale.

… Angles on a straight line add up to…?

… Angles in a quadrilateral add up to…?

… Opposite angles (on either side of a cross point) are …?…

… Angles on a straight line (about a point) add up to …?…°

… Corresponding angles (“Forresponding”) – look for the armpits of an ‘F’ formed by two parallel lines and a line crossing them (this could be back to front or upside down!)

… Alternate angles (“Zalternate”) – look for the armpits of a ‘Z’ formed by two parallel lines and a line crossing them (this could be back to front or upside down!)

… Co-interior angles (armpits of a ‘C’!) add up to …?…

… Note that for alternate, corresponding and co-interior angles you need to have two parallel lines and a line going through them.

… Equilateral triangles: all the s…?… and a…?… are the same.

… Obtuse angles are…? Acute angles are…? Reflex angles are…?

… A rhombus is a ‘squashed …?…’

… An …?… triangle has two equal sides and two equal angles.

… The angles at the base of an isosceles triangle are …?…

… What can you say about the angles in an isosceles trapezium?

… It is absolutely fine to write a calculation as a ‘reason’ in a question.

… The Rotational symmetry of an object is how many times it looks the same when you rotate it through 360°. So a rectangle has rotational symmetry of order…?

… The order of rotational symmetry of a regular polygon is equal to the number of sides.

… Forming equations by adding up the algebra terms in a triangle or quadrilateral to form an equation which you can then solve, e.g. to find ‘x’.

… If you’re not sure what to do in a question, try asking yourself, “what angles can I find using the rules I know?” This is called ‘angle chasing’. Each time you find an angle (with clear working out) add it to the diagram so that you build up a picture of the geometry.

… It can also be useful to change your perspective on the problem by turning the page upside down, sideways or covering up or highlighting part of the diagram using a ruler. This can often give you a ‘creative jump’ to help you see the next step!

… Extending lines in the diagram can help when ‘angle chasing’ as you will be able to spot alternate, corresponding and opposite angles more easily.

… If you draw a diagram, do make it nice and big!

… If you are asked to give reasons for your answers, a combination of working out calculations and words (for example, “angles on a straight line add up to…”) is a safe strategy.

Areas and Volumes

… Prisms are solids which have the same cross section if you cut through them at any point along their length. E.g. A “triangular prism”

… Volume of a prism = X-sectional Area × Length

In questions involving a prism and volume, always start by writing this formula down” you may need to rearrange it to find a length or area.

… How do you find the area of a trapezium?

… Orange juice carton and volume question. The carton is turned onto a different face and we are asked to calculate the new depth of liquid.

With these questions, try to start by asking yourself, “what CAN I find here?” The key quantity is volume, so finding the volume of liquid scores you the first mark or two.

Then redraw the new carton position using ‘d’ as the liquid depth. You can then form an equation in terms of ‘d’ as you know that the volume of liquid has remained the same.

… The area of a sector can be found by using the fraction of a whole circle method:

Area = (sector angle/360°) x πr²

… The length of a sector’s arc can be found in a similar way:

s = (sector angle/360°) x 2πr

Loci of Points

… A locus is just a fancy name for the ‘path’ of points which fit a given rule.

… Finding the locus of points which are:

a fixed distance from a given point. You would need to draw a …?…

A fixed distance from a given line

Equidistant from two given lines. This is also called an ‘angle b…?…’

Equidistant from two given points. This is also called a ‘p…?… b…?…’

… Constructing equilateral triangles (60° angles) and 30° angles using a ruler and a pair of compasses.

Loci (paths) of a point P – techniques using a pair of compasses and a ruler

… The locus (path) of points which are a fixed distance (equidistant):

- from a given point
- from a given line
- from a given shape

Constructions – using a pair of compasses

… Constructing accurate 60° angles (as part of an equilateral triangle)

… Constructing accurate 90° angles from a point on a line (use the perpendicular bisector method)

… Drawing the perpendicular from a given point to a line

… Finding the angle bisector of two lines (the locus of points which is equidistant from both lines)

… Constructing a perpendicular bisector (locus of points equidistant from two points)

You may need to use two or more of these ideas to find regions which satisfy certain conditions. For example, “shade the area which is less than 20m from the tree, but closer to the line AB than to the line AC” (for some given diagram).

Similar shapes

… To prove that two triangles are similar, show that they share exactly the same angles.

… If you see a question with a diagram (or description) of similar-looking small and large shapes… Then it’s likely to be a similar shapes question!

… Add any written information given in the question to the diagram where possible. For example, for lengths such as OB = 3OA, you could say that length OA is “1” and length OB as “3”.

… Similar triangles and shapes: using ‘Small’ and ‘Big’ column headings for matching lengths of each shape. Then work out the multiplying factor (the scale factor) to move from one column to the other. For example:

Small : Big

4cm : 10cm

L cm : 15cm

To go from the big to the small matching sides, you need to ÷10 and then ×4.

This gives us a multiplying scale factor of ×4/10, which we can now apply to the 15 length:

15 × 4/10 = 6cm

… When finding similar AREAS, the LENGTH multiplying factor is SQUARED.

… When finding similar VOLUMES, the LENGTH multiplying factor is CUBED.

… If the scale factor to go from left to right = ×4/10

then to go in the opposite direction ÷ 4/10

… If you can calculate the scale factor for VOLUME, then working up the table we can work out the scale factor for length by taking the CUBED ROOT:

Small : Big

L 1cm : ? ⇒ The scale factor for length (small to big) = ∛(8/5)

A

V 5cm³ : 8cm³ … The scale factor for volume (small to big) = ×8/5

… If you can calculate the scale factor for AREA, then working up the table we can work out the scale factor for length by taking the SQUARE ROOT:

Small : Big

L 1cm : ? ⇒ The scale factor for length (small to big) = √(4/3)

A 3cm² : 4cm² … The scale factor for area (small to big) = ×4/3

V

… Similar shapes can occur within a CIRCLE when two chords intersect. You will need to draw extra lines to create the similar triangle and then find the ‘matching’ sides (e.g. The small triangle short side matches with the big triangle’s short side).

It can be helpful to redraw each triangle separately so you can identify the ‘matching’ sides.

As the triangles are similar, the ratios of the small/big matching lengths are equal.