Transcript Angles - TheChalkface

Angles
Definitions:
Angles are a way of measuring direction or turn
One full turn is divided into 360 equal parts
One degree is
of a full turn
Why is it 360 parts?
Why not 400?
Or 6.2831853071...?
Angles around a point add up to 360°
Proof:
Turning 360° is the same as turning one
complete turn, so any angles that complete a
full turn must have a total of 360°.
Degrees are fractions of a turn.
If you turn
then
, then
of a turn,
, you will
have completed 1 full turn.
Angles on a straight line add up to 180°
Proof:
If you turn until you are pointing in the exact
opposite direction, you will have turned exactly
half of a full turn, which is half of 360°.
Remember ‘angles on a straight
line’ means at a given point, not
just any old angles that happen
to be on the same line.
Where straight lines cross,
opposite angles are equal
Proof:
Each angle is on a straight line with the one next
to it, so they add up to 180°. This means the
one to the left and the one to the right must be
the same as each other.
Crossing lines are
symmetrical, so
angles exactly
opposite each other
must be the same.
Corresponding angles on parallel lines are equal
Proof:
An angle is just a measure of direction. If two
lines cut across your line at the same angle,
they must be going in the same direction.
This is basically the
definition of parallel
lines. Another way of
saying “goes in the
same direction” is
“crosses any line at
the same angle”
Allied angles on parallel lines add up to 180°
Proof:
This comes from the corresponding angles rule.
The top angle and the one directly below must
add up to 180° because they’re on a straight
line, so the interior angles must add up to 180°
The red and blue
angles add up to 180°
(they are sometimes
called co-interior, or
supplementary or
complementary)
Alternate angles on parallel lines are equal
Proof:
Using the straight lines crossing rule we can see
that the angles on alternate sides of the
crossing line must be equal.
When a straight line
crosses parallel lines,
you only need 1 angle to
calculate all the other 7.
Angles in a triangle add up to 180°
Proof:
Angles on a straight line add up to 180° and
alternate angles on parallel lines are equal.
This is a fundamental rule that
will be extended for polygons,
so it’s important that we are
convinced it is always true.
Base angles in an isosceles triangle are equal
Proof:
An isosceles triangle has two sides the same
length. If you cut it exactly in half from the top
the two halves would be the mirror image of
each other. This means the angles at the base
of the equal sides must be equal.
Once you know one
angle in an isosceles
triangle, you can work
out the other two.
All angles in an equilateral triangle are equal
Proof:
In an equilateral triangle, all the sides are the
same length. This means it must be the same
no matter which way round it goes, so all the
angles must be the same.
Since angles in a
triangle always add up
to 180°, each angle in
an equilateral triangle
must be 60°