Parallel+and+Perpendicular+Lines

** __Marked 01/02/09__ Group Members:** Chris Walker (group leader), Kate Balinson, Talitha Brown, Aboudi Aboudi
__**Parallel and Perpendicular Lines**__

__ Definitions __
- Parallel lines __never intersect (cross)__ - The product of the slopes is -1 - Perpendicular lines __meet to form right angles (90°)__
 * Parallel -** Two lines are parallel if they have the __same slope__
 * Perpendicular -** The slopes are __negative reciprocals__ of eachother
 * Supplementary Angles-** Angles whose __sum is 180°__
 * Compelementary Angles-** Angles whose __sum is 90°__

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

= -4/-1 (Step 2: Change the numerator's sign) = 4 (Step 3: Simplify if necessary)
 * b)** -1/4 (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. ||  || ||  ||
 * c)** 0 (Step 1: Flip the number)
 * [[image:http://regentsprep.org/Regents/math/ALGEBRA/AC3/Lparal14.gif width="136" height="47"]]
 * [[image:http://regentsprep.org/Regents/math/ALGEBRA/AC3/Lparal16.gif width="113" height="47"]]

Their slopes (// m //) are negative reciprocals. (Remember //y// = // m x// + //b//.)  ||
 * [[image:http://regentsprep.org/Regents/math/ALGEBRA/AC3/Lparal18.gif width="101" height="73"]] || These lines are perpendicular.

** Example 2 Parallel lines. **     **Hint! Parallel line equations don't have exponets in them!** //y// = 3//x// - 7 //y// = 3//x// + 0.5 //y// = 3//x// ||  These lines are ALL parallel. They all have the same slope (//m//). (Remember //y// = //mx// + //b//.) || No exponents!!**__
 * //y// = 3//x// + 5

Diagrams~ __**    parallel lines :) ↑ Notice the right angle!  perpendicular lines :) ↑



Parallel lines ^

__//** 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")
 * //__ How do i construct a perpendicular line ?!

**//__ If i already have a line how do i make a line that is parallel to it ?! __//**  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.**
 * lines are parallel if they have the same slope.

__Try this!__
Here are some REAL PICTURES. What do you know... it's __MATH IN REAL LIFE__!!! (answers are below pictures)
 * -see if you can find the perpendicular lines and the parallel lines

**   Top row= Perpendicular Bottom row= parallel 
 * answers**

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:  =

And then, Supplementary and Complementary
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
 * ALTERNATE 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
 * CORRESPONDING 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°
 * CO-INTERIOR 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 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.
 * OPPOSITE ANGLES**
 * 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 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°
 * COMPLEMENTARY ANGLES**

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

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 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.

<span style="font-family: Verdana,Geneva,sans-serif;">//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