Marked 01/02/09 Group Members: Chris Walker (group leader), Kate Balinson, Talitha Brown, Aboudi Aboudi

Parallel and Perpendicular Lines

Definitions

Parallel - Two lines are parallel if they have the same slope
- Parallel lines never intersect (cross) Perpendicular - The slopes are negative reciprocals of eachother
- The product of the slopes is -1
- Perpendicular lines meet to form right angles (90°) Supplementary Angles- Angles whose sum is 180° Compelementary Angles- Angles whose sum is 90°

Example 1
Find the negative reciprocals of each of the following: a) 2 (Step 1: Flip the number)
= 1/2 (Step 2: Change the numerator's sign)
= -1/2 (Step 3: Simplify if necessary)

b) -1/4 (Step 1: Flip the number)
= -4/-1 (Step 2: Change the numerator's sign)
= 4 (Step 3: Simplify if necessary)

c) 0 (Step 1: Flip the number)
= -1/0 (Step 2: Change the numerator's sign)
= undefined *it is impossible to divide by 0.* (Step 3: Simplify if necessary) To find a negative reciprocal of a number, flip the number over (invert) and negate that value.

These lines are perpendicular.
Their slopes (m) are negative reciprocals.
(Remember y = mx + b.)

Example 2 Parallel lines. Hint! Parallel line equations don't have exponets in them!

y = 3x + 5 y = 3x - 7 y = 3x + 0.5 y = 3x

These lines are ALL parallel.
They all have the same slope (m).
(Remember y = mx + b.)

No exponents!!

See full size image

Diagrams~

parallel lines :) ↑
Notice the right angle! perpendicular lines :) ↑

Parallel lines ^

How do i construct a perpendicular line ?!

Step number 1. (red) - create a circle with centre at "P" to make points "A" and "B" on the line "AB" witch are the same distance from "P"
Step number 2. (green)- contruct circles centered at "A" and "B", both passing through "P".(Allow "Q" to be the other point of intersection for these two circles)
Step number 3. (blue)- to construct the desired line "PQ", connect "Q" and "P" (creating line "PQ")

If i already have a line how do i make a line that is parallel to it ?! lines are parallel if they have the same slope.
To create a perpendicular line to a pre-existing line you would use the formula y=mx+b!
You need to know the slope("m") to create a parallel line to a pre-existing line.

Try this!

Here are some REAL PICTURES. What do you know... it's MATH IN REAL LIFE!!! -see if you can find the perpendicular lines and the parallel lines
(answers are below pictures)

answers Top row= Perpendicular
Bottom row= parallel

How are parallel and perpendicular lines used in real life?

-architects have knowledge of parallel and perpendicular lines to enhance their drafting abilities
-carpenters understand the aspects of parallel and perpendicular lines to create perfect structures
-blacksmiths use parallel and perpendicular lines to shape and weld metal
-crafts often require the usage of parallel and perpendicular lines to achieve a desired effect
-art, specifically modern and abstract, uses the skill of parallel and perpendicular lines to convey a message
-woodworkers need to attach pieces of wood in stable, perpendicular angles
-graphing often requires the usage and identification of parallel and perpendicular lines
-cartography (map-making) needs parallel and perpendicular lines, specifically when using longitude and latitude
-product packaging requires the knowledge of parallel and perpendicular lines to understand how to create a functional, practical package
-farmers use basic knowledge of parallel lines to neatly and efficiently plant crops

Remember the Special Relations of Parallel Lines:

Alternate, Corresponding, Co-Interior, and Opposite

And then, Supplementary and Complementary

ALTERNATE ANGLES
Alternate angles are equal
Remember the Z PATTERN!

Angles C and F are alternate, and therefore equal
Angles G and B are alternate, and therefore equal

CORRESPONDING ANGLES
Corresponding angles are equal
Remember the F PATTERN or the LL PATTERN
(depending on what your teacher taught you from Gr 8)

Angles G and E are corresponding, and therefore equal
Angles C and A are corresponding, and therefore equal
Angles B and D are corresponding, and therefore equal
Angles H and F are corresponding, and therefore equal

CO-INTERIOR ANGLES
Co-interior angles add up to 180° Remember the C PATTERN

Angles B and C are co-interior, and therefore add up to 180°
Angles G and F are co-interior, and therefore add up to 180°

OPPOSITE ANGLES
Opposite angles are equal
Remember the X PATTERN

Angles H and C are opposite, and therefore equal
Angles D and G are oppostie, and therefore equal
Angles B and E are opposite, and therefore equal
Angles A and F are opposite, and therefore equal SUPPLEMENTARY ANGLES
Supplementary angles are two or more angles, that do not necessarily have to be attached, that when added together equal 180°
Any line, perpendicular or not, that intersects another line, creates four supplementary angles.

Angles A and E are supplementary, and therefore together equal 180°
Angles B and F are supplementary, and therefore together equal 180°
Angles C and G are supplementary, and therefore together equal 180°
Angles D and H are supplementary, and therefore together equal 180°

COMPLEMENTARY ANGLES
Complementary angles are to or more angles, that do not necessarily have to be attached, that when added together equal 90°
If a line at any angle shares the same point of intercect as two perpendicular lines, the angles within the right angle obviously add up to 90°

Angles I and F are complementary, and therefore together equal 90°
Angles A and J are complementary, and therefore together equal 90°

Example 1 Based on the information given, try to figure out the angle values of the variables.

How to solve it:

Identify Parallel lines within the diagram

Using the "special relations", solve for h, c, g, and f

FOR C: Discover that angle C is corresponding (F pattern) to the 30° angle, and therefore also 30°. C=30°

FOR H: Discover that angle H is opposite (X pattern) to angle C, and therefore also 30°. H=30°

FOR G: Discover that angle G is supplementary (F pattern) to angle C, (180°-30°=150°) and therefore 150°. G=150°

FOR F: Discover that angle F is co-interior (C pattern, add up to 180°) to angle G, (180°-150°=30°) and therefore 30°. F=30°

Example 2
Based on the information given, try to figure out the angle X

How to solve it:

Identify Parallel lines within the diagram

Using the "special relations" solve for x

FOR X: Notice that angle X is alternate (Z pattern) to the 50° angle, and therefore also 50°. X=50°

Example 3 Based on the information given, try to figure out the unknown values

How to solve it:

Identify Parallel lines within the diagram

Using the "special relations" solve for y, x, and z

FOR X: Discover that angle X is corresponding (F pattern) to the 79° angle, and therefore also 79°. X=79°

FOR Z: Discover that angle Z is supplementary (add up to 180°) to angle X, (180°-79°=101°) and therefore 101°. Z=101°

Citations

Hendriks, Justyn. "Parallel and Perpendicular Lines." Ancaster, Canada. Nov. 2008.

Hendriks, Justyn. "Angles and Parallel Lines." Ancaster, Canada. 30 May 2008.

Math Power 9 Ontario Edition. (1999). Toronto: McGraw-Hill Ryerson.

Chris Walker (group leader), Kate Balinson, Talitha Brown, Aboudi AboudiMarked 01/02/09Group Members:Parallel and Perpendicular LinesDefinitionsParallel -Two lines are parallel if they have thesame slope- Parallel lines

never intersect (cross)Perpendicular -The slopes arenegative reciprocalsof eachother- The product of the slopes is -1

- Perpendicular lines

meet to form right angles (90°)Supplementary Angles-Angles whosesum is 180°Compelementary Angles-Angles whosesum is 90°Example 1Find the negative reciprocals of each of the following:

a)2 (Step 1: Flip the number)= 1/2 (Step 2: Change the numerator's sign)

= -1/2 (Step 3: Simplify if necessary)

b)-1/4 (Step 1: Flip the number)= -4/-1 (Step 2: Change the numerator's sign)

= 4 (Step 3: Simplify if necessary)

c)0 (Step 1: Flip the number)= -1/0 (Step 2: Change the numerator's sign)

=

undefined(Step 3: Simplify if necessary)*it is impossible to divide by 0.*To find a negative reciprocal of a number, flip the number over (invert) and negate that value.

Their slopes (

m) are negative reciprocals.(Remember

y=mx+b.)Example 2 Parallel lines.Hint! Parallel line equations don't have exponets in them!y= 3x+ 5y= 3x- 7y= 3x+ 0.5y= 3xThey all have the same slope (

m).(Remember

y=mx+b.)Diagrams~

parallel lines :) ↑

Notice the right angle!

perpendicular lines :) ↑

Parallel lines ^

How do i construct a perpendicular line ?!

Step number 1. (

red) - create a circle with centre at "P" to make points "A" and "B" on the line "AB" witch are the same distance from "P"Step number 2. (

green)- contruct circles centered at "A" and "B", both passing through "P".(Allow "Q" to be the other point of intersection for these two circles)Step number 3. (

blue)- to construct the desired line "PQ", connect "Q" and "P" (creating line "PQ")If i already have a line how do i make a line that is parallel to it ?!lines are parallel if they have the same slope.To create a perpendicular line to a pre-existing line you would use the formula y=mx+b!

You need to know the slope("m") to create a parallel line to a pre-existing line.

Try this!Here are some REAL PICTURES. What do you know... it's

MATH IN REAL LIFE!!!-see if you can find the perpendicular lines and the parallel lines

(answers are below pictures)

answersTop row= Perpendicular

Bottom row= parallel

## How are parallel and perpendicular lines used in

?real life-architects have knowledge of parallel and perpendicular lines to enhance their drafting abilities

-carpenters understand the aspects of parallel and perpendicular lines to create perfect structures

-blacksmiths use parallel and perpendicular lines to shape and weld metal

-crafts often require the usage of parallel and perpendicular lines to achieve a desired effect

-art, specifically modern and abstract, uses the skill of parallel and perpendicular lines to convey a message

-woodworkers need to attach pieces of wood in stable, perpendicular angles

-graphing often requires the usage and identification of parallel and perpendicular lines

-cartography (map-making) needs parallel and perpendicular lines, specifically when using longitude and latitude

-product packaging requires the knowledge of parallel and perpendicular lines to understand how to create a functional, practical package

-farmers use basic knowledge of parallel lines to neatly and efficiently plant crops

## Remember the Special Relations of Parallel Lines:

## Alternate, Corresponding, Co-Interior, and Opposite

## And then, Supplementary and Complementary

ALTERNATE ANGLESAlternate angles are

theequalRemember

Z PATTERN!

Angles C and F are alternate, and therefore equal

Angles G and B are alternate, and therefore equal

CORRESPONDING ANGLESCorresponding angles are

theequalRemember

F PATTERNor theLL PATTERN(depending on what your teacher taught you from Gr 8)

Angles G and E are corresponding, and therefore equal

Angles C and A are corresponding, and therefore equal

Angles B and D are corresponding, and therefore equal

Angles H and F are corresponding, and therefore equal

CO-INTERIOR ANGLESCo-interior angles

add up to 180°RemembertheC PATTERNAngles B and C are co-interior, and therefore add up to 180°

Angles G and F are co-interior, and therefore add up to 180°

OPPOSITE ANGLESOpposite angles are

theequalRemember

X PATTERNAngles H and C are opposite, and therefore equal

Angles D and G are oppostie, and therefore equal

Angles B and E are opposite, and therefore equal

Angles A and F are opposite, and therefore equal

SUPPLEMENTARY ANGLESSupplementary angles are two or more angles, that do not necessarily have to be attached, that when added together equal 180°

Any line, perpendicular or not, that intersects another line, creates four supplementary angles.

Angles A and E are supplementary, and therefore together equal 180°

Angles B and F are supplementary, and therefore together equal 180°

Angles C and G are supplementary, and therefore together equal 180°

Angles D and H are supplementary, and therefore together equal 180°

COMPLEMENTARY ANGLESComplementary angles are to or more angles, that do not necessarily have to be attached, that when added together equal 90°

If a line at any angle shares the same

point of intercectas two perpendicular lines, the angles within the right angle obviously add up to 90°Angles I and F are complementary, and therefore together equal 90°

Angles A and J are complementary, and therefore together equal 90°

Example 1Based on the information given, try to figure out the angle values of the variables.

How to solve it:

C=30°H=30°G=150°F=30°Example 2Based on the information given, try to figure out the angle X

How to solve it:

X=50°Example 3Based on the information given, try to figure out the unknown values

How to solve it:

X=79°Z=101°CitationsHendriks, Justyn. "Parallel and Perpendicular Lines." Ancaster, Canada. Nov. 2008.

Hendriks, Justyn. "Angles and Parallel Lines." Ancaster, Canada. 30 May 2008.

Math Power 9 Ontario Edition. (1999). Toronto: McGraw-Hill Ryerson.Pierce, Rod. "Parallel Lines" Math Is Fun. Ed. Rod Pierce. 11 Sep 2007. 19 Dec 2008

http://www.mathsisfun.com/geometry/parallel-lines.html

Pierce, Rod. "Supplementary Angles" Math Is Fun. Ed. Rod Pierce. 14 Jul 2006. 19 Dec 2008

http://www.mathsisfun.com/geometry/supplementary-angles.html

Pierce, Rod. "Complementary Angles" Math Is Fun. Ed. Rod Pierce. 15 May 2007. 19 Dec 2008

http://www.mathsisfun.com/geometry/complementary-angles.html

Google. 18 Dec. 2008http://images.google.ca/imghp?hl=en&tab=wi