Goals: In this unit you will learn about how angles are measured and created for workplace application and how combinations of angles and lines can be used to create parallel and non-parallel lines. You will use your mathematical skills and knowledge to: o Measure, draw and describe angles o Estimate the measure of angles o Use certain angles to determine whether 2 lines are parallel o Solve problems involving angles, pairs of angles, parallel, non-parallel, perpendicular and transversal lines Key Terms: Angle Angle measure Degree Parallel lines Perpendicular lines transversal
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Apprenticeship and workplace Math 10
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Unit 5.1:Measuring, Drawing and Estimating Angles. Angle: Two arms that meet at a point called the vertex. Angles are measured in degrees using tools such as a protractor. arm
Vertex
arm
True Bearing: The angle measured clockwise between North and an intended path or direction expressed in degrees Bearings have 3 digits 090 or 145 000 is regarded as North. Angle Measure: The number of degrees between two arms joined at the vertex. Angle Referent: A common standard of angle measure eg. 90 , 45 , 30 and 22.5 used to estimate angles Drawing Angles: You need the following tools to draw certain angles. Set square Ruler Compass.
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Apprenticeship and workplace Math 10
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Example 1. Use a ruler and compass to create the following angles a) Draw a 90 angle b) Replicate any existing angle. Solution: a) Follow these steps to draw a 90 angle Draw a line segment and mark where you want your 90 to go Put your compass point at your 90 mark and open slightly. Make 2 arcs at equal distance on either side of your 90 mark Widen the compass a bit more and place the point at one of the new marks and make a small arc. Repeat at the other new mark so the 2 small arcs intersect. Draw a line segment that goes through the 1st mark you made and the intersecting arcs.
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Apprenticeship and workplace Math 10
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b) Follow these steps to replicate any existing angle. Use a compass centered at the vertex of the original angle to draw an arc through both arms. Use a ruler to draw one arm of the new angle. Draw an arc of the same radius and arc length as the one you just drew on the original angle. Take the compass to the original angle and set it so that its point and tip of the pencil touch the points where the original arc intersects the sides of the angle. Place the compass point at the intersection of the new arc and arm and draw a small arc through the new one. Use the ruler to draw the other arm through the intersecting arcs.
Example 2. Estimate the measure of the angle in the diagram .
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Solution:
Example 3. Estimate the measure of this angle below:
Solution:
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Referents diagrams: Shows angles from 0o to 360 A clock face is a good reference diagram.
Complementary angles: 2 angles that add up to 90 Supplementary angles: 2 angles that add up to 180
Example 4. Sort the following into pairs of complementary and supplementary angles: 42
121
107 31
59
19
48
73
Solution:
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Apprenticeship and workplace Math 10
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Compass Rose N
W
E
S The angles in a compass rose are a fraction of 360 and measured in a clockwise direction from the North line. Example 5. a) Determine the true bearing between A and B N B
A
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b) Determine the bearing of B to A N
B A
Solution: a) Estimate first using the compass rose as e referent and then measure the angle using a protractor.
The bearing from A to B is
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b) Measure the acute angle and subtract from 360
Complete notebook assignment
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page 184 # 1-6
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Apprenticeship and workplace Math 10
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Unit 5.2: Angle Bisectors and Perpendicular lines. Angle Bisectors: A segment, arm or line that separates 2 halves of a bisected line. 72 is bisected into two 36 angles
36
Bisector
36
Example 1. Accurately bisect an angle like the one shown here.
Solution:
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Do Activity 5.4 on page 190
Complete notebook assignment
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page 192 # 1-7
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Apprenticeship and workplace Math 10
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Unit 5.3 Non-Parallel Lines and Transversals Vertically Opposite Angles: angles created by intersecting lines that share only a vertex.
1
2
3
4
Transversal: A line that intersects 2 or more lines.
L1 transversal L2
Corresponding Angles: Angles that occupy the same relative position in 2 different sections
1
2
3 5 7
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4 6 8
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Apprenticeship and workplace Math 10
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Alternate Interior Angles: Angles in opposite positions between 2 lines intersected by a transversal
1
2
3
5 7
4
6 8
Interior Angles: Interior Angles on the same side of a transversal
1
2
3
5 7
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4
6 8
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Apprenticeship and workplace Math 10
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Alternate Exterior Angles: Angles in opposite positions outside 2 lines intersected by a transversal
1
2
3
5 7
4
6 8
Exterior Angles: Exterior Angles on the same side of a transversal.
1
2
3
5 7
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4
6 8
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Example 1. Below is the side diagram of porch that is attached to a house.
Roof joist
3 2
1
column
Wall
5
4
floor
For each pair of angles below identify: i) the kind of angle pair ii) the parts of the porch that make up the angle pairs a) 1 and 4
b) 3 and 5
c) 1 and 3
Solution: a) i)____________________________________________________ _____________________________________________________ ii)___________________________________________________ _____________________________________________________ S.Duffy
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b) i)____________________________________________________ _____________________________________________________ ii)___________________________________________________ _____________________________________________________
c) i)____________________________________________________ _____________________________________________________ ii)___________________________________________________ _____________________________________________________
Example 2. An electricity pylon is made up of pairs of line segments and transversals. The diagram below shows part of a pylon.
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Cross brace 1 Cross brace 3
1
2
Cross brace 2
3 5
4 6
The following list contains 3 pairs of angles and the type that each angle pair is. Determine which 2 parts of the tower make up the main line segments, and which part that makes up the transversals that form each of these pairs of angles.
a) 3 and 4 are corresponding angles
b) 2 and 5 are alternate interior angles
c) 1 and 6 are exterior angles on the same side of the transversal
Unit 5.4: Parallel Lines and Transversals. When a transversal cuts through 2 parallel lines Corresponding angles are equal : 1=5; 2=6; 3=7; 4=8
1
2
3
5 7
4
6 8
Alternate interior angles are equal: 3=6;
1
2
3
5 7
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4=5
4
6 8
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Apprenticeship and workplace Math 10
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Alternate exterior angles are equal:
1
1=8; 2=7
2
3
4
5
6
7
8
Interior angles on the same side of a transversal add up to 180
:
3+5=180
1
2
3
5 7
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4+6=180
4
6 8
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Apprenticeship and workplace Math 10
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Exterior angles on the same side of a transversal add up to 180 : 1+7=180
1
2
3
5 7
2+8=180
4
6 8
Example 1. Calculate the missing angles Solution:
76
b
a
c
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Example 2. Calculate the missing angles. Solution:
x
Complete notebook assignment S.Duffy
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page 214 # 1-7 Page 22
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Reflect on your learning Now that you have completed this unit check
the box that applies
to you
RED
AMBER
GREEN
I understand all the key terms.
I can draw, measure and Describe angles of various measures I can determine whether pairs of angles are complementary or supplementary I can use referents to estimate angles. I can bisect angles
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I can identify and calculate corresponding, alternate interior and alternate exterior angles around transversals I can determine whether lines are parallel or not I have completed all homework assignments. I have attended lunchtime tutorials for extra help.
