Download Angles

Algebra A
Lines and Angles
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Lines
In Mathematics, a straight line is defined as having infinite
length and no width.
Is this possible in real life?
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Labelling line segments
When a line has end points we say that it has finite length.
It is called a line segment.
We usually label the end points with capital letters.
For example, this line segment
A
B
has end points A and B.
We can call this line ‘line segment AB’.
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Labelling angles
When two lines meet at a point an angle is formed.
A
B
C
An angle is a measure of the rotation of one of the line
segments relative to the other.
We label points using capital letters.
The angle can then be described as
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ABC or
CBA.
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Lines in a plane
What can you say about these pairs of lines?
These lines cross,
or intersect.
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These lines do
not intersect.
They are parallel.
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Lines in a plane
A flat two-dimensional surface is called a plane.
Any two straight lines in a plane either intersect once …
This is called
the point of
intersection.
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Lines in a plane
… or they are parallel.
We use arrow
heads to show
that lines are
parallel.
Parallel lines will never meet.
They stay an equal distance apart.
We can say that parallel lines are always equidistant.
Where do you see parallel lines in everyday life?
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Perpendicular lines
What is special about the angles at
the point of intersection here?
a=b=c=d
a
b
d
c
Each angle is 90. We show
this with a small square in
each corner.
Lines that intersect at right angles are called
perpendicular lines.
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Angles
Angles are measured in degrees.
A quarter turn
measures 90°.
90°
It is called a right
angle.
We label a right
angle with a small
square.
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Angles
Angles are measured in degrees.
A half turn measures
180°.
180°
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This is a straight line.
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Angles
Angles are measured in degrees.
A three-quarter turn
measures 270°.
270°
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Angles
Angles are measured in degrees.
360°
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A full turn measures
360°.
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You must learn facts about angles.
So you can calculate their size without drawing or measuring.
•
•
•
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Learn facts about
Angles between intersecting lines
Angles on a straight line
Angles around a point
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Vertically opposite angles
When two lines intersect, two pairs of vertically opposite
angles are formed.
a
d
b
c
a=c
and
b=d
Vertically opposite angles are equal.
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Angles on a straight line
Angles on a line add up to 180.
a
b
a + b = 180°
because there are 180° in a half turn.
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Angles around a point
Angles around a point add up to 360.
a
b
c
d
a + b + c + d = 360
because there are 360 in a full turn.
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Calculating angles around a point
Use geometrical reasoning to find the size of the labelled
angles.
69°
a
167°
103°
68°
d
c
43°
b
43°
137°
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Complementary angles
When two angles add up to 90° they are called
complementary angles.
a
b
a + b = 90°
Angle a and angle b are complementary angles.
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Supplementary angles
When two angles add up to 180° they are called
supplementary angles.
a
b
a + b = 180°
Angle a and angle b are supplementary angles.
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Angles made with parallel lines
When a straight line crosses two parallel lines eight
angles are formed.
a
b
d
c
e
f
h
g
Which angles are equal to each other?
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Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
d = h because
Corresponding angles are equal
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Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
a = e because
Corresponding angles are equal
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Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
c = g because
Corresponding angles are equal
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Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
b = f because
Corresponding angles are equal
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Alternate angles
There are two pairs of alternate angles, or Z-angles.
a
b
d
c
e
f
h
g
d = f because
Alternate angles are equal
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Alternate angles
There are two pairs of alternate angles, or Z-angles.
a
b
d
c
e
f
h
g
c = e because
Alternate angles are equal
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Angles in a triangle
c
a
b
For any triangle,
a + b + c = 180°
The angles in a triangle add up to 180°.
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Calculating angles in a triangle
Calculate the size of the missing angles in each of the
following triangles.
116°
a
33°
31°
b
64°
326°
82°
49°
43°
d
25°
c
88°
28°
233°
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Angles in an isosceles triangle
In an isosceles triangle, two of the sides are equal.
We indicate the equal sides by drawing dashes on them.
The two angles at the bottom of the equal sides are called
base angles.
The two base angles are also equal.
If we are told one angle in an isosceles triangle we can work
out the other two.
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Angles in an isosceles triangle
For example,
88°
a
46°
a
46°
Find the sizes of the
other two angles.
The two unknown angles are equal so call them both a.
We can use the fact that the angles in a triangle add up to
180° to write an equation.
88° + a + a = 180°
88° + 2a = 180°
2a = 92°
a = 46°
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Interior angles in triangles
The angles inside a triangle are called interior angles.
b
c
a
The sum of the interior angles of a triangle is 180°.
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