Presentation on theme: "What do all of these have in common?. Proving Two Lines are Parallel n k m 3 2 1 Given: m || n, n || k Prove: m || k StatementsReasons 1.m || nGiven 2.<1."— Presentation transcript:
Proving Two Lines are Parallel n k m 3 2 1 Given: m || n, n || k Prove: m || k StatementsReasons 1.m || nGiven 2.<1 = <2Corresponding Angles Postulate 3.n || kGiven 4.<2 = < 3Corresponding Angles Postulate 5.<1 = < 3Transitive Property of Congruence 6.m || kCorresponding Angles Postulate ~ ~ ~
Theorems about Parallel & Perpendicular Lines Theorem 3.11 If two lines are parallel to the same line, then they are parallel to each other. Theorem 3.12 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. p q r If p || q and q || r, then p || r. m n p If m _|_ p and n _|_ p, then m || n.
Explaining why steps are parallel and building a CD Rack In this picture, each step is parallel to the step immediately below it, and the bottom step is parallel to the floor. Why is the top step parallel to the floor? You are building a CD rack or book shelf. You cut the sides, bottom, and top so that each corner is composed of two 45* angles. How can you prove that the top and bottom of the rack are parallel?
Finding the Slope of a Line If a cog railway goes up the side of a mountain at a rate of 4 feet for each 10 feet it goes forward. What is the slope of this section of the railroad? Rise Slope = --------- = ------- = Run From Algebra we learned that the slope of a line is equal to it’s “rise” over it’s “run.” If the line passes through the points (X1, y1) and (X2, Y2), then the slope is determined by: Rise Y2 – Y1 Slope = --------- = ------------ = Run X2 – X1 * Slope is generally represented by the variable m
Using Slope to Determine if Lines are Parallel Postulate 17: 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 j k If the line j has a rise of 4 and a run of 2, and line k has a rise of 2 and a run of 1, are the two lines parallel?
Identifying Parallel Lines (-4,0) (0,1) (2,0) (-2,6) (0,6) (-6,5) j k l Find the slope of each line. j = k = l = Which lines are parallel?
Writing the equation for a line Again, in algebra we learned that we can use the slope (m) of a non-vertical line to write an equation for the line in slope-intercept form. y = mx + b, where m = slope of the line, and b = y - intercept How would we write an equation for a line that passes through the point (2,3) and has a slope of 5? First: Solve for b. (use (x,y) – (2,3) and m = 5 y = mx + b Then write the equation for the line.
Writing an Equation for a Parallel Line (3, 2) n p Line p has the equation y = -1/3x – 1 Line n is parallel to p and passes through the point (3,2). Write and equation for line n. 1.Find the slope of n. 2. Solve for b using the coordinate information given. 3. Write an equation for y =, in slope intercept form
Parallel Lines in the Coordinate Plane – Practical Applications A zip line is a taunt rope or cable that you can ride down on a pulley. If a zip line goes from a 90 foot tall tower to a 60 foot tall tower, and the towers are 200 feet apart, what is the slope of the zip line? What do you think would be important factors in planning a zip line?
Parallel Lines in the Coordinate Plane – Civil Engineering The warning signs posted by the US DOT particularly benefit drivers of large trucks. In cases where long steep grades end at busy intersections, runaway truck ramps can be constructed parallel to the road. A driver who anticipates or experiences difficulty from mechanical failures such as failed brakes, can exit onto the ramp and stop more safely. The slope of the road is called the road’s grade. Grades are measured in percents. For example, if the slope of a road is 1/20, the grade is 5%. A warning sign is required for any hill that fits one of the following descriptions: 5% grade and more than 3000 feet long 6% grade and more than 2000 feet long 7% grade and more than 1000 feet long 8% grade and more than 750 feet long 9% grade and more than 500 feet long What is the grade of the hill to the nearest %, and is a sign needed? The hill is 1400 feet long and drops 70 feet. The hill is 2200 feet long and drops 140 feet. The hill is 600 feet long and drops 55 feet. The hill is 450 feet long and drops 40 feet.