Angle Theorems

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Oct 10, 2013 (3 years and 6 months ago)

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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

L
ines 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 opposi
te).


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 s
ometimes 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, th
e two angles
labelled y are equal.

We know that x, y and z

t
ogether 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 q
uadrilateral, 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 yo
u 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 poin
t:


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 a
re
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 S
ame 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
circumfe
rence 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 circumferen
ce 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 an
d 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
ci
rcle'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 ab
le 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 sho
rt, 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
relate
d 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 = πr
2

× 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