9 3 Parallel Lines in a Coordinate Plane

9 -3 Parallel Lines in a Coordinate Plane Geometry

Standard/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: Find slopes of lines and use slope to identify parallel lines in a coordinate plane. � Write equations of parallel lines in a coordinate plane.

Slope of parallel lines � In algebra, you learned that the slope of a nonvertical line is the ratio of the vertical change (rise) to the horizontal change (run). � If the line passes through the points (x 1, y 1) and (x 2, y 2), then the slope is given by slope = rise run m = y 2 – y 1 x 2 – x Slope is usually represented by the variable m.

Finding the slope of train tracks � COG RAILWAY. The cog railway goes up the side of Mount Washington, the tallest mountain in New England. At the steepest section, the train goes up about 4 feet for each 10 feet it goes forward. What is the slope of this section? slope = rise = 4 feet =. 4 run 10 feet

Finding Slope of a line Find the slope of the line that passes throug the points (0, 6) and (5, 2). m = y 2 – y 1 x 2 – x 1 =2– 6 5– 0 =-4 5

Postulate: Slopes of Parallel Lines � In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. Lines k 1 and k 2 have the same slope. k 1 k 2

Deciding whether lines are parallel � Find the slope of each line to determine if l 1║l 2? M 1 = 4 =2 2 M 2 = 2 1 =2 Because the lines have the same slope, l 1║l 2.

Identifying Parallel Lines M 1= 0 -6 = -3 2 -0 2 M 2 = 1 -6 = -5 0 -(-2) 0+2 2 M 3 = 0 -5 = -5 -4 -(-6) -4+6 2 l 3 l 2 l 1

Solution Compare the slopes. Because k 2 and k 3 have the same slope, they are parallel. Line k 1 has a different slope, so it is not parallel to either of the other lines.

Writing Equations of parallel lines In algebra, you learned that you can use the slope m of a non-vertical line to write an equation of the line in slopeintercept form. slope y-intercept y = mx + b The y-intercept is the y-coordinate of the point where the line crosses the y-axis.

Writing an Equation of a Line Write an equation of ln through the point (2, 3) that has a slope of 5. Steps/Reasons why y = mx + b Slope-Intercept form 3 = 5(2) + b Substitute 2 for x, 3 for 3 = 10 + b -7= b y and 5 for m Simplify Subtract.

Write the equation � Because is m = - 1/3 and b = 3, the equation of ln y = -1/3 x + 3

HOMEWORK p. 345 #4 -12 (mo 4) plus 13, 16 �

The Triangle Sum Theorem Section 9 -4

Parallel postulate Given a line and a point not on the line, there is one and only one line through the point parallel to the line. Draw as many lines as possible through the point, and make them all parallel to the given line.

Cut out triangle experiment Cut out the triangle Draw little arrows pointing to the vertices On a sheet of paper, draw a large triangle. Rip off the corners!

Cut out triangle experiment Repeat Draw Take one a line with ofand the another put corner and pieces a point Once of onmore. your near the middle. triangle and set it on the line so the arrow points to the dot.

Triangle sum theorem � The total measure of the angles of a triangle is 180° � When you know two angles of a triangle, add them together then subtract from 180 to get the third angle.

88 88 +57 145 x 57 X = 35 180 -145 35

x 41 63 X= a) 104 b) 76 c) 24

83 29 x X= a) 112 b) 61 c) 68

HOMEWORK p. 352 #4 -32 (mo 4)