# Parallel lines

Categories: gcse geometry

Here are some basic geometry rules about parallel lines. They are also covered in this video:

## Angles around intersection lines

When two lines intersect, four angles are a formed:

The angles around intersecting lines form two pairs of *opposite and equal* angles. The two angles labelled *a* are equal, and the two angles labelled *b* are equal. This means that there are only two different angles at the intersection.

Since angles *a* and *b* form a straight line, we also know that:

```
a + b = 180°
```

## Angles around parallel lines

If we draw an extra line, parallel to the previous line, it looks like this:

The small arrows on the two horizontal lines indicate that they are parallel with each other.

Since the lines are parallel, they both make the same angle with the line that crosses them. So both intersections only involve the two angles *a* and *b*. This means that if we know the value of any angle, we can find out all the other angles. For example, if we are told that *a* is 120°, we know that all the angles labelled *a* in the diagram are 120°. We can then calculate that *b* is 60° (since *a + b* is 180°), so we know all the angles.

There are several common cases of matching pairs of angles around pairs of angles. These are given special names - alternate angles, allied angles, and corresponding angles.

## Alternate angles

If you take one of the angles from the first intersection, and the opposite angle from the second intersection, they are called alternate angles:

Alternate angles are equal. In this case the angles are both equal to *b*.

The diagram shows that you can spot alternate angles by a characteristic Z shape. But the shape might be rotated or flipped so it might not always look exactly like a Z. Here is an example of a different pair of alternate angles:

## Allied angles

You could think of the intersection as being one half of a parallelogram:

Allied angles are the two "internal" angles in that half of the parallelogram:

Allied angles add up to 180°. We can see this because there will always be one *a* and one *b* angle.

Allied angles are sometimes called *co-linear angles* or *internal angles*.

The diagram shows that you can spot allied angles by a characteristic C shape. The shape might be rotated or flipped so it might not always look exactly like a C. Here is an example of a different pair of allied angles:

## Corresponding angles

If you take one of the angles from the first intersection, and the equivalent angle from the second intersection, they are called corresponding angles:

Corresponding angles are equal. In this case the angles are both equal to *a*.

The diagram shows that you can spot corresponding angles by a characteristic F shape. Again, the shape might be rotated or flipped so it might not always look exactly like an F. Here is an example of a different pair of corresponding angles:

## Join the GraphicMaths Newletter

Sign up using this form to receive an email when new content is added:

## Popular tags

adjacency matrix alu and gate angle area argand diagram binary maths cartesian equation chain rule chord circle cofactor combinations complex polygon complex power complex root cosh cosine cosine rule cpu cube decagon demorgans law derivative determinant diagonal directrix dodecagon ellipse equilateral triangle eulers formula exponent exponential exterior angle first principles flip-flop focus gabriels horn gradient graph hendecagon heptagon hexagon horizontal hyperbola hyperbolic function infinity integration by substitution interior angle inverse hyperbolic function inverse matrix irregular polygon isosceles trapezium isosceles triangle kite koch curve l system locus maclaurin series major axis matrix matrix algebra minor axis nand gate newton raphson method nonagon nor gate normal not gate octagon or gate parabola parallelogram parametric equation pentagon perimeter permutations polar coordinates polynomial power product rule pythagoras proof quadrilateral radians radius rectangle regular polygon rhombus root set set-reset flip-flop sine sine rule sinh sloping lines solving equations solving triangles square standard curves star polygon straight line graphs surface of revolution symmetry tangent tanh transformations trapezium triangle turtle graphics vertical volume of revolution xnor gate xor gate