Angles

There are a number of Basic Rules governing Angles. You need to know them all.

Angles on a straight line add up to 180° a + b + c = 180°

Angles around a point add up to 360° a + b + c + d + e = 360°

Angles in a triangle add up to 180° a + b + c = 180°

Exterior Angle of a Triangle = Sum of Opposite Interior Angles d = a + c

Angles in a quadrilateral add up to 360° a + b + c + d = 360°

Learn all the facts above.

Three angles in a quadrilateral are 70°, 120° and 84°. What is the fourth angle?

The sum of the angles of a quadrilateral is 360° so the missing angle is: 360° - (70° + 120° + 84°) = 86°

The sum of the angles in a triangle is 180°

a + b + c = 180°

In an equilateral triangle, all the angles are 60°

The angles between each equal side and the unequal side in an isosceles triangle are the same.

You only need to know one angle in an isosceles triangle to be able to work out the other two.

Find a, b in the diagram.

The angles between each equal side and the unequal side in an isosceles triangle are the same. So, a = 70° The angles in a triangle add up to 180° So, b = 180° - (70° + 70°) = 40°

If two lines intersect, then the Opposite Angles are Equal

Learn the diagram

What is the size of angle a?

It's opposite 115° so it's equal to 115°

Angles around a point add up to 360°

a + b + c + d + e = 360°

No matter how many angles there are around the point, they add up to 360°

What is the size of x in the diagram?

Angles around a point add up to 360° So, x = 360° - (47° + 90° + 62°) = 161°

The Interior Angles between a pair of parallel lines are Supplementary ie. they add up to 180°

Note that the Interior Angles have to be on the same side of the line joining the parallel lines.

Look for the C or U shape to spot interior angles.

What is the size of x in the diagram below?

The lines are parallel so x and 118° are interior angles (do you see the C shape?) So, x = 180° - 118° = 62°

Alternate Angles lie within parallel lines on opposite sides of the transversal that crosses the line. They are equal

Look for the Z shape.

What is the size of x in the diagram below?

x is alternate to the angle that's 119° so, x = 119°

Corresponding Angles lie in the same relative position between parallel lines and the transversal crossing them. They are equal.

Look for the F shape

Which angle 'corresponds' to x and so is equal?

In this example, the F shape is upside down. x and c 'correspond' because they lie in the same relative position regarding their parallel lines and the transversal. So, x = c

Angles in a quadrilateral add up to 360°

a + b + c + d = 360°

Look out for special quadrilaterals, especially those which have one or more pairs of parallel sides.

What's the size of x in the quadrilateral below?

The quadrilateral is a trapezium because it has one pair of parallel sides.

By Interior Angles, (a + 130°) = 180° So a = 50° And 2a = 100°

Again by Interior Angles, (x + 100°) = 180° so, x = 80°

A Regular Polygon is a Polygon with equal sides, equal interior angles and equal exterior angles.

For any regular polygon: Exterior Angle = 360° ÷ (number of sides) Interior Angle = 180° - Exterior Angle

Let x be the exterior angle of a regular polygon (a heptagon is shown but the same proof applies to any regular polygon).

Because it's a regular polygon, DB = BA, and CD = CB = CA (where C is the centre of the polygon). So, triangles BCD and ACB are identical and are isosceles with equal angles x.

Now, the exterior angle of a triangle = the sum of the opposite interior angles so, for triangle ABC, (e+x) = (x+y) So, e = y

Let N be the number of sides of the polygon. At the centre, Ny = 360° And there are N exterior angles, so Exterior Angle = 360°/N

And, as angles on a straight line add up to 180°, each Interior Angle = 180° - Exterior Angle

Remember the formula for the Exterior and Interior Angles of a Regular Polygon and don't get them muddled up.

A Regular Polygon has internal angles that are 157.5°. How many sides does it have?

Interior Angle = 180° - Exterior Angle So, Exterior Angle = 180° - 157.5° = 22.5° Now, for a regular polygon, Exterior Angle = 360° ÷ (number of sides) So, (number of sides) = 360° ÷ Exterior Angle = 360° ÷ 22.5° = 16