Geometry Unit 3 Equations of Lines/Parallel & Perpendicular Lines

Transcription

1 Geometry Unit 3 Equations of Lines/Parallel & Perpendicular Lines Lesson Parallel Lines & Transversals Angles & Parallel Lines Slopes of Lines Assignment 174(14, 15, 20-37, 44) 181(11-19, 25, 27) *TYPO in student edition book #27 Worksheet Quiz Review Quiz Equations of Lines Proving Lines Parallel Perpendiculars & Distance 200(13, 16, 18, 19, 23, 25, 26, 29, 31, 32, 39, 40, 43-47) (2, 4, 8-15, 16, 18, 20, 33, 34) 219(13, 14, 15, 18, 23, 32) Review Test VOCABULARY Parallel Skew Interior Angles Exterior Angles Consecutive Interior Alternate Interior Alternate Exterior Corresponding Slope Perpendicular Transversal Distance Slope Intercept Form Point Slope Form

2 Geometry Parallel Lines & Transversals An Ames room creates the illusion that a person standing in the right corner is much larger than a person standing in the left corner. (See identical twins, right.) From a viewing hole, the front and back walls appear parallel, when in fact they are slanted. The ceiling and floor appear horizontal, but are actually tilted. The construction of the Ames room above makes use of intersecting, parallel, and skew lines, as well as intersecting and parallel planes, to create an optical illusion. Parallel & Skew Parallel lines are coplanar lines that do not intersect. J K Example: JK LM L M Arrows are used to indicate that lines are parallel. Skew lines are lines that do not intersect and are not coplanar. Example: Lines l and m are skew. l A Parallel planes are planes that do not intersect. Example: Planes A and B are parallel. B m

3 A line that intersects two or more coplanar lines at two different points is called a transversal. Transversal Angle Pair Relationships Four interior angles lie in the region between lines q and r. Four exterior angles lie in the two regions that are not between lines q and r. Consecutive interior angles are interior angles that lie on the same side of the transversal. Alternate interior angles are nonadjacent interior angles that lie on opposite sides of transversal t. Alternate exterior angles are nonadjacent exterior angles that lie on opposite sides of transversal t. Corresponding angles lie on the same side of transversal t and on the same side of lines q and r. 3, 4, 5, 6 1, 2, 7, 8 4 and 5, 3 and 6 3 and 5, 4 and 6 1 and 7, 2 and 8 1 and 6, 2 and 5, 3 and 8, 4 and 7

4

5 Geometry Angles & Parallel Lines Corresponding Angles Postulate If two parallel lines are cut by a transversal, them each pair of corresponding angles is congruent. Examples: Example: In the figure, m 8 = 105. Find the measure of each angle. Tell which postulate or theorem you used. t m 1 = m 2 = 1 2 n 4 3 m 3 = 8 7 p m 4 = 5 6 m 5 = m 6 = m 7 = Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. Examples: Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary. Examples: Alternate Exterior Angles Theorem If two lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

6 Perpendicular Transversal Theorem In a plane, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other. t a Example: If line a b and line a line t, then line b line t. b

7 Geometry Slopes of Lines Ski resorts assign ratings to their ski trails according to their difficulty. A primary factor in determining this rating is a 6 trail s steepness or slope gradient. A trail with a 6% or 100 grade falls 6 feet vertically for every 100 feet traveled horizontally. The easiest trails have slopes ranging from 6% to 25%, while more difficult trails have slopes of 40% or greater. vertical rise Slope of a Line horizontal run y y m = = 2 y1 x x2 x1 ( means or ) Example: Find the slope of each line. 1A) 1B) the line containing (6, -2) and (-3, -5) the line containing (8, -3) and (-6, -2) 1C) 1D) the line containing (4, 2) and (4, -3) the line containing (-3, 3) and (4, 3) Classifying Slopes Positive Slope Negative Slope Zero Slope Undefined slope

8 Slope can be interpreted as a rate of change, describing how a quantity y changes in relationship to quantity x. The slope of a line can also be used to identify the coordinates of any point on the line. Example: DOWNLOADS In 2006, 500 million songs were legally downloaded from the Internet. In 2004, 200 million songs were legally downloaded. A) Use the data given to graph the line that models the number of songs legally downloaded y as a function of time x in years. B) Find the slope of the line and interpret its meaning. C) If this trend continues at the same rate, how many songs will be legally downloaded in 2010? Parallel & Perpendicular Lines You can use the slopes of two lines to determine whether the lines are parallel or perpendicular. Lines with the same slope are. Lines with slopes that are negative reciprocals of each other are. Example: Determine whether AB and CD are parallel, perpendicular or neither. A) A(14, 13), B(-11, 0), C(-3, 7), D(-4, -5) B) A(3, 6), B(-9, 2), C(5, 4), D(2, 3)

9

10

11 Geometry Equations of a Line Slope-intercept form m= b= Point slope form m= (x 1, y 1 )= Ex 1: Write an equation in slope intercept form if the slope is 3 and y-intercept is -2. Graph the line. Ex 2: Write an equation in point slope form if m=-3/4 that contains (-2,5). Graph the line.

12 Ex 3: Write an equation of a line that contains (0, 3) and (-2, -1) in slope intercept form. Ex 4: Write an equation of a line that contains (-2, 6) and (5, 6) in slope intercept form. **Key Concept** X=a is a vertical line Y=b is a horizontal ine Ex 5: Write an equation in slope intercept form for a line perpendicular to the line y=-3x+2 and goes through (4, 0).

13 Geometry Proving Lines Parallel Postulate 3.4 If corresponding angles are congruent then the lines are parallel. Ex Theorems Summary: Ex 1 Determine if the lines are parallel given the following information: A < 1 = < 6 B < 2 = < 3 Ex 2 Find m<mrq so that a // b. Show your work:

14

15 Geometry Perpendiculars and Distance Equidistant: Ex 1 Construct the segment that represents the distance indicated. Ex 2 Line l contains points at (-5, 3) and (4, -6). Find the distance between line l and point (2, 4). ( Do ALL WORK on GRAPH PAPER ). Ex 3 Find the distance between parallel lines l and m if their equations are y=2x+1 and y=2x-4 respectively.

16

17

18

19