Parallel lines will never meet, no matter how far they are extended. Here are some interesting facts about angles and parallel lines.

Parallel lines

In addition to the basic angle facts, you need to memorise the rules of angles and parallel lines, and apply these to problems.

Small arrows are used to show that 2 lines are parallel to each other.

Click on the button below to see examples of these different types of angle.

• Vertically opposite angles are equal.
• Corresponding angles are equal.
• Alternate angles are equal.
• Co-interior angles add up to 180°.

Maths

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Question

Find the angles a, b and c. Give reasons for your answers.

Question

Find the angle B A D. Give a reason for your answer.

When answering questions on angles, remember the following points:

• Look out for parallel lines (squares, rectangles, parallelograms and marked parallel lines) and see which of the above rules apply.
• Look out for lines of equal length. Remember that an isosceles triangle has two sides of equal length and that the angles which are directly opposite these sides are also equal.
• Diagrams are unlikely to be drawn to scale, so do not be tempted to measure!
• Do not assume things which may turn out to be untrue (for example, an angle is not necessarily a right angle just because it looks as though it should be!).
• It is likely that you will not be able to find the required angle immediately. Do not worry - just write as much information as possible onto the diagram. It should eventually lead you to the answer. If not, you might still get some marks for your working!

Solving simultaneous equations using graphs

Solving simultaneous equations using a graph is easier than you might think. First, you need to draw the lines of the equations. The points where the lines cross is the solution.

Linear equations

The graphs of linear equations will give straight lines.

Example

• Solve these simultaneous equations by drawing graphs:
• 2x + 3y = 6
• 4x - 6y = - 4

For example, to draw the line 2x + 3y = 6 pick two easy numbers to plot. One when x = 0 and one where y= 0

• When x = 0 in the equation 2x + 3y = 6
• This means 3y = 6 so y = 2
• So one point on the line is (0, 2)

• When y = 0
• 2x = 6 so x = 3
• So another point on the line is (3 ,0)

In an exam, only use this method if you are prompted to by a question. It is usually quicker to use algebra if you are not asked to use graphs.

Linear and quadratic equation

Example

• Solve the simultaneous equations by drawing graphs.
• y - 2x = 1
• y = x² - 2

How to convert a recurring decimal into a fraction

Example 1

Write 0.77..$\mathrm{0.77..}$ as a fraction in its lowest terms.

Our first step is to form a simple equation where x=0.77..$x=\mathrm{0.77..}$. By multiplying both sides by 10$10$ we can obtain another equation with 10x=7.77..$10x=\mathrm{7.77..}$. Now we eliminate the recurring part of the decimal by subtracting x$x$ from 10x$10x$.

x10x9xx=0.77..=7.77..=7=79$$\begin{array}{rl}x& =\mathrm{0.77..}\\ 10x& =\mathrm{7.77..}\\ 9x& =7\\ x& =\frac{7}{9}\end{array}$$

So we have our answer 0.77...=79$\mathrm{0.77...}=\frac{7}{9}$.
The important part to remember is to get two equations in x$x$ where the recurring part after the decimal point is exactly the same.

Example 2

Write 0.277..$\mathrm{0.277..}$ as a fraction in its lowest terms.

Again our first step is write x=0.277..$x=\mathrm{0.277..}$. Now multiplying by 10$10$ gives us 10x=2.77..$10x=\mathrm{2.77..}$. This time we need another equation to match the recurring part of the equation. So multiplying by 10$10$ again gives 100x=27.77..$100x=\mathrm{27.77..}$. Now we have two equations with the same recurring part we subtract one from the other as before.

x10x100x90xxx=0.277..=2.77..=27.77..=25=2590=518$$\begin{array}{rl}x& =\mathrm{0.277..}\\ 10x& =\mathrm{2.77..}\\ 100x& =\mathrm{27.77..}\\ 90x& =25\\ x& =\frac{25}{90}\\ x& =\frac{5}{18}\end{array}$$

So in its lowest terms 0.277..=518$\mathrm{0.277..}=\frac{5}{18}$.

Example 3

Write 0.0855.. as a fraction in its lowest terms.

As always our first step it to write x=0.0855..$x=\mathrm{0.0855..}$. This time to get two equations with the same recurring part after the decimal point we need to multiply by 100$100$ and then 1000$1000$. This gives us:

100x1000x900xx=8.55..=85.55..=77=77900$$\begin{array}{rl}100x& =\mathrm{8.55..}\\ 1000x& =\mathrm{85.55..}\\ 900x& =77\\ x& =\frac{77}{900}\end{array}$$

This cannot be simplified any further so 0.855..=77900$\mathrm{0.855..}=\frac{77}{900}$.

Example 4

Write 0.0234234.. as a fraction in its lowest terms.

This time the '234$234$' part is the important bit. Now to get two multiples of x with 234 as the recurring part we need to multiply first by 10$10$ and then another 1000$1000$ to move the digits 3 places to the right.

x=0.0234234...10x=0.234234..10000x=234.2349990x=234xx=2349990x=13555$$\begin{array}{r}x=\mathrm{0.0234234...}\\ 10x=\mathrm{0.234234..}\\ 10000x=234.234\\ 9990x=234x\\ x=\frac{234}{9990}\\ x=\frac{13}{555}\end{array}$$

So 0.0234=13555$0.0234=\frac{13}{555}$ as a fraction in its lowest terms.