Section 8.1, 8.2, 8.3, 8.4
Parallel Lines – are lines on a flat surface that never meet.
Ex: opposite side of a rectangular picture frame
railroad/roller coaster tracks
lines on a sheet of loose-leaf paper
guitar strings
Parallel lines are always the same distance apart and are represented using the following symbols.
A B this means
Or line AB is parallel to
C D line CD.
Take two minutes to list as many pairs of parallel lines in the classroom as you can.
Constructing Parallel Lines
● An easy way to construct/draw parallel lines is to use both sides of a ruler.
Questions:
1. Identify pairs of parallel lines from each shape.
A B E F
C D G H
I J M N
K L O P
Q R W
V S X Y
U T Z
2. Construct a line that is parallel to each line below.
Y
A). Q B).
X
P
C). Draw a line segment EF of length 5 cm.
Draw a line parallel to EF.
D). How can you check to see if the lines are actually parallel and not that they just look
parallel?
Perpendicular Lines – two linesegments are perpendicular if they meet at right angles
(900)
Ex: wall meeting the floor : a cross
: fence posts and fence rails : four way stop
Constructing Perpendicular Lines
Method #1 - draw a straight line
- place the base of the plastic right triangle from your geometry set along the
line.
- draw a line along the height of the right triangle.
- use your protractor to make sure it is 900.
Method #2 - draw a straight line
- mark a dot at the center.
- place the center of your protractor on the dot.
- measure 900 and mark another dot
- join the two points.
- use your protractor to measure and make sure it is 900.
Symbol for perpendicular C
A B
D
`
Questions:
1. Can two lines be both parallel and perpendicular?
● Impossible. Parallel lines never meet and perpendicular lines meet at 900.
2. Can a line have more than one line perpendicular to it?
● Yes, like a railway track or a ladder.
3. Construct a line perpendicular to each line below.
C
A). B B).
D
A
C). Draw a line segment EF of length 5 cm.
Draw a line perpendicular to EF.
D). How can you check to see if the lines are actually perpendicular and not that they look
perpendicular?
Perpendicular Bisector
To bisect a line means to divide a line into two equal parts.
A C B
A perpendicular bisector is a line that intersects another line at a right angle and divides it into two equal parts.
D
and
A C B
=
E
Constructing Perpendicular Bisectors
Method #1 - draw a line
- fold your paper so that A lands on B.
- the fold line is the perpendicular bisector.
Method #2 - Use your ruler and draw a line of 10cm.
A B
- place the ruler so that A is on one side of the ruler and B is on the other.
- Draw lines along each edge of the ruler.
A
B
- repeat this step with A and B on opposite sides of the ruler.
Draw the lines along the edge of the ruler.
A B
- label the points, C and D, where the lines you sketched intersected.
C
A E B
D
- is the perpendicular bisector of
- use your protractor to check if its 900. Call this point E.
- measure = . They should be 5 cm each.
Practice Questions
Draw a perpendicular bisector for a line that is:
A). 8 cmB). 5 cm
Method #3 - use a ruler and a protractor
- Draw a line AB = 10cm.
- Label the midpoint C.
- Place the center of your protractor at C and mark off 900, label it D.
- is the perpendicular bisector of
D
A C B
Definitions
Acute angle Right Angle
– an angle measuring less than 900. - an angle that is exactly 900.
Obtuse AngleStraight Angle
- an angle that is more than 900 - is a straight line.
but less than 1800. It measures exactly 1800.
Reflex Angle
- is more than 1800, but less than 3600.
It’s easiest to subtract the acute angle from 3600, the answer is the reflex angle.
Constructing Angle Bisectors
For every angle there exists a line that divides the angle into two equal parts. This line is called the angle bisector.
Draw an angle of any size with your ruler.
Using your compass draw an arc from the vertex B. Label the intersection points A and C as shown.
A
B
C
Keep your compass opened at the same size. Place your compass on A and draw another arc towards the center of the angle. Do the same from point C. Where the new arcs overlap, call point D. Connect B and D, this is the angle bisector. Verify using your protractor.
A
B
Examples: Construct an angle bisector for each angle below.
A). Obtuse angleB). Acute angle
Constructing an Angle of a Certain Size
1. Draw a 1260 angle.
- draw a straight line, mark a dot at the start of the line.
- place the center of your protractor on the dot and measure off 1260.
- connect the new point with the dot on your line. This angle is 1260.
1260
Examples:
1. Draw an angle that is:
A). 840 B). 1000
2. Draw an angle bisector for each angle 1260, 840 and 1000.
.
Section 8.5: Graphing on a Coordinate Grid
A vertical number line and a horizontal number line that intersect at right angles at zero form a coordinate grid or coordinate plane.
The horizontal axis is the x-axis. (It goes across).
The vertical axis is the y-axis. (It goes up and down).
The axes meet at the origin (0,0) and divide the plane into four quadrants. Thequadrants
are numbered counter clockwise.
y-axis
21
x-axis
origin
3 4
An ordered pair is a point with two coordinates (x, y).
The first number represents the horizontal direction. Left is negative, right is positive.
The second number represents the vertical direction. Down is negative, up is positive.
Both numbers are directions to get you to one point.
Always start at (0,0) and follow the directions of the coordinates.
Examples:
1. Plot the points A(1,-3) , B(-4, 2),
C (3, -2) , D(-2,-6) , E(-6, 0) , F (0,4)
and determine which quadrant
they are located.
2. Write the coordinates of each point.
Answers:
3. Plot the following points.
P(-3, 0) Q (-4, -5 ) R (-1, 1) S (3, 5)
4. In which quadrant would you find each point?
A). Q (-3,-3)
B). Z (2, -3 )
C). B (1, 4)
**** When necessary you can change the scale on the x and y axis of a coordinate grid.
Instead of always increasing by 1, you can increase by 2, 5 , 10 etc. Be consistent!
5. Choose a scale and draw a coordinate grid. Plot each point.
A (-20, -15) B(10, -15) C (-20, 20)
- Due to the large numbers, it would make sense to use a scale of 5, instead of 1.