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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.
An angle is a measure of the rotation of one of the line segments relative to the other.
We label points using capital letters.
A
BC
The angle can then be described as 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.
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.
Parallel lines will never meet. They stay an equal distance apart.
Where do you see parallel lines in everyday life?
We can say that parallel lines are always equidistant.
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Perpendicular lines
What is special about the angles at the point of intersection here?
a = b = c = d
Lines that intersect at right angles are called perpendicular lines.
Lines that intersect at right angles are called perpendicular lines.
ab
cd Each angle is 90. We show
this with a small square in each corner.
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Angles
Angles are measured in degrees.
A quarter turn measures 90°.
It is called a right angle.
We label a right angle with a small square.
90°
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Angles
Angles are measured in degrees.
A half turn measures 180°.
This is a straight line.180°
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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.
A full turn measures 360°.360°
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You must learn facts about angles.So you can calculate their size without drawing or measuring.
• 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
b
c
d
a = c and b = d
Vertically opposite angles are equal.Vertically opposite angles are equal.
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Angles on a straight line
Angles on a line add up to 180.Angles on a line add up to 180.
a + b = 180°
ab
because there are 180° in a half turn.
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Angles around a point
Angles around a point add up to 360.Angles around a point add up to 360.
a + b + c + d = 360
a b
cd
because there are 360 in a full turn.
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b c
d
43° 43°
68°
Calculating angles around a point
Use geometrical reasoning to find the size of the labelled angles.
103°
a167°
137°
69°
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Complementary angles
When two angles add up to 90° they are called complementary angles.
When two angles add up to 90° they are called complementary angles.
ab
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.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.
Which angles are equal to each other?
ab
c
d
ef
g
h
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dd
hh
ab
ce
f
g
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
ab
ce
f
g
d = h because
Corresponding angles are equalCorresponding angles are equal
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ee
aab
c
d
f
g
h
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
b
c
d
f
g
h
a = e because
Corresponding angles are equalCorresponding angles are equal
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gg
cc
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
c = g because
ab d
ef h
Corresponding angles are equalCorresponding angles are equal
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ff
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
b = f because
ab
c
d
e
g
h
b
Corresponding angles are equalCorresponding angles are equal
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ff
dd
Alternate angles
There are two pairs of alternate angles, or Z-angles.
d = f because
Alternate angles are equalAlternate angles are equal
ab
ce
g
h
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ccee
Alternate angles
There are two pairs of alternate angles, or Z-angles.
c = e because
ab
g
h
d
f
Alternate angles are equalAlternate angles are equal
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Angles in a triangle
For any triangle,
a b
c
a + b + c = 180°
The angles in a triangle add up to 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.
233°
82°31°
116°
326°
43°49°
28°
ab
c
d
33°64°
88°
25°
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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,
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°
a
a
88° + 2a = 180°2a = 92°a = 46°
46°
46°
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Interior angles in triangles
c a
b
The angles inside a triangle are called interior angles.
The sum of the interior angles of a triangle is 180°.The sum of the interior angles of a triangle is 180°.