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.

external image Lparal14.gif

external image Lparal15.gif

external image Lparal16.gif

external image Lparal17.gif

external image Lparal18.gif
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
See full size image


Diagrams~

###_a_paralellllinne.gif

parallel lines :) ↑
###_a_perpendiculaaaeeeliiiiningjfgs.gif Notice the right angle! external image 600px-Arrow_northwest.svg.png
perpendicular lines :) ↑

external image 600px-Parallel_Lines.svg.png

Parallel lines ^

How do i construct a perpendicular line ?!
###_construction_of_perpendicular_lines.png

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)

external image perplines.jpg external image sidewalk.jpgexternal image d90_rack1.JPGexternal image 2298571313_a2fb2568f7.jpg%3Fv%3D1204245282
external image 42572743_9fb8ce5864.jpgexternal image feld.JPGexternal image ParallelLines.jpgexternal image 295390944_edad92b738.jpg%3Fv%3D0



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!
parallel_lines_alternate.jpg

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)
parallel_lines_corresponding.jpg
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
parallel_lines_co-interior.jpg
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
parallel_lines_opposite.jpg
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.

parallel_lines_supplementary.jpg
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°
parallel_lines_complementary.jpg
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.
parallel_lines_example_1.jpg
How to solve it:
  1. Identify Parallel lines within the diagram
  2. Using the "special relations", solve for h, c, g, and f
  3. FOR C: Discover that angle C is corresponding (F pattern) to the 30° angle, and therefore also 30°. C=30°
  4. FOR H: Discover that angle H is opposite (X pattern) to angle C, and therefore also 30°. H=30°
  5. FOR G: Discover that angle G is supplementary (F pattern) to angle C, (180°-30°=150°) and therefore 150°. G=150°
  6. 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

parallel_lines_example_2.jpg
How to solve it:
  1. Identify Parallel lines within the diagram
  2. Using the "special relations" solve for x
  3. 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
parallel_lines_example_3.2.jpg
How to solve it:

  1. Identify Parallel lines within the diagram
  2. Using the "special relations" solve for y, x, and z
  3. FOR X: Discover that angle X is corresponding (F pattern) to the 79° angle, and therefore also 79°. X=79°
  4. 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.

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