Blog Entry #4

The Converse of the Corresponding Angles Postulate is used for constructing parallel lines. The Parallel Postulate swears that for any line, you can construct a parallel line through a point that is not on the line.

Following postulates and theorems, can help prove lines parallel, and prove angles congruent.
Other properties may apply to lines and could be used to prove lines parallel.

 Page 168 #42, the question asks if the information that is given allows you to conclude that the two lines A and B are parallel.

     Yes, they are parallel and the given information allows you to conclude that.

     Line A is intersected by a line (vertical angles.) The angle measures 125 degrees and one of these pairs are its opposite. Vertical angles have equal measures, so that means that the measure of the opposite angle 125 = 125 degrees. The vertical angle of angle 125 and angle 55 are considered same-side interior angles since they lie on the same side of the transversal between  lines A and B. Same-side interior angles are supplementary, they equal 180 when added (125 + 55 = 180) By using the Converse of the Same-Side Interior Angles Theorem, we can conclude that line A is parallel to line B.



     Page177 #27,( I drew this out but my camera isn’t working) In my drawing, segment AB is a perpendicular bisector of segment XY BUT segment XY is not a perpendicular bisector of segment AB.

    Page 177 #28 The rungs of the ladder are parallel because the two rungs are perpendicular to the same side. Theorem 3-4-3 states that, "If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other." By using this, it’s safe to say that the two coplanar lines are the rungs and the line is the side of the ladder. This means that the rungs are parallel.
    

    To find the slope of a line, you can use any two points on the line.

     If line W has a slope of a/b, then the slope of a line perpendicular to line W is -b/a. Two nonvertical lines are perpendicular iff the product of their slopes is -1. (Vertical and horizontal lines are perpendicular.

     Page 186 #23, the question asks for an inequality for the slope of a line perpendicular to line AB, while line AB has a slope greater than 0 and less than 1.

    By looking at the problem, the slope of a line greater than 0 would be m > 0. The slope of a line less than one would be m < 1. Since the problem gives two values in one sentence, we form a compound inequality. So the slope of line AB would be 1 > m > 0

     

     When perpendicular lines are multiplied, the product formed is -1. So this means that if the slope of line AB is 1 > m > 0, the slope of a line perpendicular to line AB should be its negative response. 1(x) = -1.The new slope should include -1 > m. 0(x)= -1 results with x = -1/0, a.k.a. undefined. After taking out undefined, the slope of a line perpendicular to line AB would be -1 > m. The slope of a line perpendicular to line AB is m < -1

     Page 186 #24) asks, if two cars are driving at the same speed, what is true about the lines that represent the distance traveled by each car?

   Both cars should pass the same distance with the time, if both cars are travelling at the same speed. The slope represents the speed. Because the speeds are the same, the slopes should be the same. When following the Parallel Line Theorem, we can conclude that the two lines that represent the distance are parallel, since the two lines have the same slope, and lines with the same slope are parallel lines.

     What are two ways to determine the slope of a line?

     One way to find the slope of a line is by using the slope formula. The slope formula is m = y2 - y1 / x2 - x1. To find y2, y1, x2, and x1, you can use any two points on the line. After choosing two points, put them into the formula and solve. The result is the slope of the line.

     Another way to find the slope of a line is by using a graph. When looking at the graph choose any two points. From there we can determine the slope using the ratio of rise to run. The rise is the difference in the y-values of any two points on the line, while the run is the difference in x-values. Find the rise by counting how many spaces the point was moved vertically.Up is positive, down is negative. After finding the rise, find the run. Count the spaces horizontally until you get to your second point. Left is negative, right is positive. The ratio of rise to run should be expressed as a fraction or whole number. This is the slope.

      Draw graphs of pairs of parallel lines and perpendicular lines and explain how their slopes are related.

Parallel Lines:

   As you can see in the photo, parallel lines don't intersect. 

They extend forever and never cross. For them to never cross, they would need to have the same slope, or else they would end up crossing. Because of this, parallel lines have the same slope. 

 

Perpendicular Lines:

     The slopes of perpendicular lines are negative reciprocals. When multiplied their product is -1.