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
the two tangents (from the point where they touch the
where they cross) will be the same.
cross, the lengths of
circle to the point
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.
triangle AOX is therefore isosceles and so <OXA = a
The
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