Angle Theorems

Angle Theorems

Angles are measured in degrees, written °. The maximum angle is 360°. This is the angle all the way round a point. Half of this is the angle on a straight line, which is 180°.

Related Angles

Lines AB and CD are parallel to one another (hence the » on the lines).
a and d are known as vertically opposite angles. Vertically opposite angles are equal. (b and c, e and h, f and g are also vertically opposite).
g and c are corresponding angles. Corresponding angles are equal. (h and d, f and b, e and a are also corresponding).
d and e are alternate angles. Alternate angles are equal. (c and f are also alternate). Alternate angles form a 'Z' shape and are sometimes called 'Z angles'.
a and b are adjacent angles. Adjacent angles add up to 180 degrees. (d and c, c and a, d and b, f and e, e and g, h and g, h and f are also adjacent).
d and f are interior angles. These add up to 180 degrees (e and c are also interior).
Any two angles that add up to 180 degrees are known as supplementary angles.

Angle Sum of a Triangle

Using some of the above results, we can prove that the sum of the three angles inside any triangle always add up to 180 degrees. If we have a triangle, you can always draw two parallel lines like this:

Now, we know that alternate angles are equal. Therefore the two angles labelled x are equal. Also, the two angles labelled y are equal. We know that x, y and z together add up to 180 degrees, because these together is just the angle around the straight line. So the three angles in the triangle must add up to 180 degrees.

Angle Sum of a Quadrilateral

A quadrilateral is a shape with 4 sides.

Now that we know the sum of the angles in a triangle, we can work out the sum of the angles in a quadrilateral.

For any quadrilateral, we can draw a diagonal line to divide it into two triangles. Each triangle has an angle sum of 180 degrees. Therefore the total angle sum of the quadrilateral is 360 degrees.

Exterior Angles

The exterior angles of a shape are the angles you get if you extend the sides. The exterior angles of a hexagon are shown:

A polygon is a shape with straight sides. All of the exterior angles of a polygon add up to 360°. because if you put them all together they form the angle all the way round a point:

Therefore if you have a regular polygon (in other words, where all the sides are the same length and all the angles are the same), each of the exterior angles will have size 360 ÷ the number of sides. So, for example, each of the exterior angles of a hexagon are 360/6 = 60°.

Interior Angles

The interior angles of a shape are the angles inside it. If you know the size of an exterior angle, you can work out the size of the interior angle next to it, because they will add up to 180° (since together they are the angle on a straight line).

Exterior Angle of a Triangle

Angle x is an exterior angle of the triangle:

The exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices. In other words, x = a + b in the diagram.

Proof:
The angles in the triangle add up to 180 degrees. So a + b + y = 180.
The angles on a straight line add up to 180 degrees. So x + y = 180.
Therefore y = 180 - x. Putting this into the first equation gives us: a + b + 180 - x = 180. Therefore a + b = x after rearranging. This is what we wanted to prove.

Circle Theorems

Circles

A circle is a set of points which are all a certain distance from a fixed point known as the centre.
A line joining the centre of a circle to any of the points on the circle is known as a radius.

The circumference of a circle is the length of the circle. The circumference of a circle = 2 × π × the radius.

The red line in the second diagram is called a chord. It divides the circle into a major segment and a minor segment.

Theorems

Angles Subtended on the Same Arc

Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.

Angle in a Semi-Circle

Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.

Proof

We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.

But all of these angles together must add up to 180°, since they are the angles of the original big triangle.

Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.

Tangents

A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it). A tangent to a circle forms a right angle with the circle's radius, at the point of contact of the tangent.

Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.

Angle at the Centre

The angle formed at the centre of the circle by lines originating from two points on the circle's circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.

Proof

You might have to be able to prove this fact:

OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so OXA = a
Similarly, OXB = b

Since the angles in a triangle add up to 180, we know that XOA = 180 - 2a
Similarly, ∠BOX = 180 - 2b
Since the angles around a point add up to 360, we have that AOB = 360 - ∠XOA - ∠BOX
= 360 - (180 - 2a) - (180 - 2b)
= 2a + 2b = 2(a + b) = 2 AXB

Alternate Segment Theorem

This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.

Proof

You may have to be able to prove the alternate segment theorem:

We use facts about related angles:

A tangent makes an angle of 90 degrees with the radius of a circle, so we know that OAC + x = 90.
The angle in a semi-circle is 90, so BCA = 90.
The angles in a triangle add up to 180, so BCA + OAC + y = 180
Therefore 90 + OAC + y = 180 and so OAC + y = 90
But OAC + x = 90, so OAC + x = OAC + y
Hence x = y

Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.

Area of Sector and Arc Length

If the radius of the circle is r,
Area of sector = πr2 × A/360
Arc length = 2πr × A/360
In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360