Revision:Angles

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This section of revision notes will illustrate facts about angles between lines and in polygons.

Names for Angles

• Acute angles. An angle of less than 90 degrees is called an acute angle. Angle a is acute.
• Right angle. An angle of exactly 90 degrees is called a right angle. We mark then with a small square. Angle b is a right angle.
• Obtuse angle. An angle greater than 90 degrees and less than 180 degrees is called an obtuse angle. Angle c is obtuse.
• Reflex angle. An angle greater than 180 degrees is called a reflex angle. Angle d is reflex.

Basic Angles Facts

Supplementary Angles

Any two angles that add up to 180 degrees are known as supplementary angles.

Angles on a Straight Line

Angles on a straight line add up to 180 degrees.

Angles at a Point

The angles at a point add up to 360 degrees.

Parallel line angles

Lines AB and CD are parallel to one another.

Vertically Opposite Angles

Vertically opposite angles are always equal.

In the main diagram, angles a and d are vertically opposite. So are angles b and c, angles e and h and angles f and g.

Corresponding Angles

Corresponding angles are always equal.

In the main diagram, angles b and f are corresponding angles. So are angles a and e, angles c and g and angles d and h.

Corresponding angles are also sometimes called 'F-angles' because of the F-shape they make.

Alternate Angles

Alternate angles are always equal.

In the main diagram, angles d and e are alternate angles. So are angles c and f.

Alternate angles are sometimes called 'Z-angles' because of the Z-shape they make.

Adjacent Angles

Adjacent angles always add up to 180 degrees.

In the diagram angle a and b are adjacent angles. There are lots of pairs of adjacent angles (a and c, b and d, c and d, e and g, e and f, f and h, g and h are all adjacent angles).

Adjacent angles are situated next to each other.

Interior or Allied Angles

Interior angles (also called allied or inner angles) always add up to 180 degrees.

In the main diagram, angles c and e are interior angles. So are angles d and f.

Interior angles are sometimes called 'C-angles' or 'U-angles' because of the u or c shape they make.

Angles in polygons

A polygon is a many sided shape the simplest being a triangle with just three sides. A polygon has interior angles which are angles inside the shape and exterior angles which are angles created outside the shape by extending their sides (see the diagram below).

Interior Angles in Triangles

The interior angles in a triangle add up to 180 degrees.

Interior Angles in Quadrilaterals

The angles in a quadrilateral add up to 360 degrees.

Interior angles in any polygon

The interior angles in an n-sided polygon will add up to 180(n - 2) degrees.

For example, the pentagon below the interior angles will add up to 180(5 - 2) = 180 x 3 = 540 degrees.

In regular polygon (where all the sides are of equal length), all the interior angles are equal. We can find an interior angle of a regular polygon by finding the total they all add up to using the above method and then dividing that total by the number of sides.

For example, the interior angles in a regular pentagon will add up to 540 degrees. There are 5 sides to a pentagon. Therefore, one interior angle is 540 divided by 5, which gives 108 degrees.

Exterior angles in any polygon

The exterior angles of any polygon add up to 360 degrees.

The diagram shows the exterior angles of a pentagon, which will add up to 360 degrees.

In a regular polygon, all exterior angles are equal.

For example one exterior angle of a regular pentagon is 360 divided by 5, which gives 72 degrees.

The Interior And Exterior Angle At Any Corner

At any corner of a polygon, the interior and exterior angles are supplementary. That is, they add up to 180 degrees.

Exam Tips

• Some angle questions will ask for reasons for your answers. For these questions you need to say which rule you used to work out the answer (such as angles in a triangle add up to 180 degrees or the angles are alternate).

• Do not try to measure angles in questions as the diagrams will not be drawn accurately.

• There may be more than one way to work out some questions. If you are stuck doing the question one way see if you can find another way to answer it.

• You may have to find some unlabelled angles first in order to work out the labelled ones the question asks for.

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