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Chapter 5
All about angles
Section 5.1 Types of angles
Summary of angles types:
What is an angle?
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Angle: formed by two line segments that start from the same point, which is
called a vertex.
Typically angles are measured in degrees. (Other methods do exist)
the symbol used to show degrees is
o
Measuring angles
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The tool used to measure angles is called a protractor.
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A protractor can be used to find any angle
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If the angle is larger than 180 o than more than one measurement is
required.
How to use a protractor
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Make sure the zero indicator is on the base line of the angle you want to
measure.
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Read the scale on the protractor and see how many degrees the angle is.
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Most protractors come with a double scale on them so they work from both
directions.
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It is important not to read the wrong scale.
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The scale you want to read should start at the zero.
Example 1
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What is the measure of the following angle?
Example 2
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Find the measure of the following angle to the nearest degree.
Example 3
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Find the measure of the following angle to the nearest degree.
Practice problems
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Worksheet on finding angles
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Page 226 # 7 identifying types of angles
5.2 Angle constructions
By using a protractor it is possible to make sure the angle you are making is
accurate.
The protractors we are using only measure up to 180 o
If you want to create a larger reflex angle you need to make sure you indicate
the direction of measure.
Lets create an angle of 30 degrees
Lets create an angle of 65 degrees
Create an angle of 200 degrees
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Make sure you show the direction of measure.
Bisecting angles
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Bisect: Means to cut in half
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Angle bisector: A line which cuts an angle into two equal pieces.
When we bisect angles we have to show the angle bisector.
Once you determine the angle of the bisector you still need to draw it properly.
Bisecting angles
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Bisect the following angle
This is a right angle (90º).
To bisect the angle we need to know
what half of a 90º angle is.
90º ÷ 2 = 45º
Draw a line meeting the vertex that is at
45º.
Drawing an angle bisector
Practice problems
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Worksheet on drawing angles and bisecting angles
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P. 238 # 1-7
5.3 Lines and Angles
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When two lines meet an angle is always created. At this point we have only
looked at single angles where two lines meet.
Multiple Lines & Angles
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If two or more lines cross paths then more than a single angle is created.
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Look at the example below
Many angles are created by these three lines crossing paths
Parallel lines
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Parallel lines: Two lines which never pass over each other.
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In math, they are often marked with matching arrowheads to indicate they
are parallel to remove any question of being parallel.
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It is important to remember that two parallel lines are always the same
distance apart. If they were getting farther/closer they would eventually
cross paths.
Examples of parallel lines
Perpendicular lines
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Lines which cross paths at right angles. These types of lines always make
angles of 90⁰.
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The intersection of the two lines is usually marked with a right angle symbol.
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Any intersection of lines with this symbol are perpendicular (90⁰)
Transversal lines
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A line that passes over two or more parallel lines is called a “transversal line”
Multiple Angle Relationships
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Complementary angles: Two angles that add together to
give 90⁰. This means they add up to give a right angle.
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The right angle indicator is not always shown for this relationship.
Multiple Angle Relationships cont.
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Supplementary angles: Two angles that add together to
give a total of 180 ⁰.
This means a straight angle is created when you have supplementary angles.
Practice problems
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Page 250 # 1-5
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Worksheet on complementary angles
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Worksheet on supplementary angles
Opposite Angles
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A pair of equal angles that form when two lines cross.
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Opposite angles are easy to spot as they make an “X” pattern.
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In this diagram w = y and x = z
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Opposite angles are always equal in this situation.
Corresponding Angles
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A pair of angles on the same side of a transversal crossing parallel lines.
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Often described as making an “F” pattern.
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Corresponding angles are always equal.
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Both angles indicated with “O” would be the same!
Same Side Exterior Angles “O”
Same Side Interior Angles “X”
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Same side exterior: A pair of angles on the same side of the transversal and
outside the parallel lines. They always add up to 180 ⁰. (see “O”)
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Same side interior: A pair of angles on the same side of a transversal and on
the inside of the parallel lines. They always add up to 180 ⁰. (see “X”)
Alternate Exterior Angles “O”
Alternate Interior Angles “X”
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Alternate Exterior Angles: A pair of angles on the opposite sides of a
transversal and outside the parallel lines. They are always equal. (see “O”)
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Alternate Interior Angles: A pair of angles on the opposite sides of a
transversal and inside the parallel lines. They are always equal. (see “X”)
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Usually found when a “Z” pattern exists between the angles.
Practice with various angles
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P.254 # 1-5
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Work sheets
1.Opposite angles
2. Corresponding angles
3. Alternate angles
Section 5.4 Applications of Angles
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Assigned work
p. 266 # 1-6
p. 270 # 1-6
p. 272 # 2, 3, 4, 6, 7, 8, 9