day 1

Day 1:  The objective is to review; slopes of parallel and perpendicular lines.



Parallel lines and their slopes are easy.  Slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope, and lines with the same slope are parallel.
Perpendicular lines are a more complex. If you see a line with positive slope , then the perpendicular line must have negative slope . So perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that means, you take the one slope value, and turn it upside down. Put this together with the sign change, and you get that the slope of the perpendicular line is the "negative reciprocal" of the slope of the 1st line — and two lines with slopes that are negative reciprocals of each other are perpendicular to each other. In numbers, if the one line's slope ism = ⁴/₅, then the perpendicular line's slope will be m = ^–5/₄. If the one line's slope is m = –2, then the perpendicular line's slope will be m = ¹/₂.
I Here's how it works:

• One of the lines passes through the points (–1, –2) and (1, 2); another line passes through the points (–2, 0) and (0, 4). Are these lines parallel, perpendicular, or neither?

To answer this question, We will find the slopes.

Since these two lines have the same slopes, then the lines are parallel.

• One line passes through the points (0, –4) and (–1, –7); and   another line passes through the points (3, 0) and (–3, 2). Are these lines parallel, perpendicular, or neither?

We will find the values of the slopes. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

 
If w were to flip the "3" and then change its sign, We would get " ^–1/₃". In other words, these slopes are negative reciprocals, so the lines through the points are perpendicular.

• One line passes through the points (–4, 2) and (0, 3); another line passes through the points (–3, –2) and (3, 2). Are these lines parallel, perpendicular, or neither?

We will find the slopes.

 
These slope values are not the same, so the lines are not parallel. The slope values are not negative reciprocals either, so the lines are not perpendicular. Then the answer is "neither".


1. Problem: Find the measure of each numbered angle in the figure below. Given: Line GH is parallel to ray DK Angle 6 = 75 degrees. Angle 2 = 30 degrees. Solution: Angle 5 = 105 degrees since it is supplementary to Angle 6. Angle 4 = 45 degrees because of the rule outlined above. (Angle 4 + angle 2 = angle 6, so angle 4 = angle 6 - angle 2.) Angle 1 = 45 degrees since angles 1 and 4 are alternate interior angles. Angle 3 = 105 degrees since angles 3 and 5 are alternate interior angles.

2.  If two lines are cut by a tranversal, and the corresponding angles are congruent (congruent angles have the same measure), the lines are parallel.  Example:

      Problem: If angles 2 and 3 are congruent,
      are lines r and s parallel?
  
       Solution: Angle 2 = angle 3 - Given.
               Angle 1 = angle 2 - Vertical angles
                                   are congruent.
               Angle 1 = angle 3 - Transitive Property:
                                   If a = b and b = c, 
                                   then a = c.
               r is parallel to s by the above
               rule.

3. If two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary, the lines are parallel.  Example:

    1.  Problem: Show that lines a and b
                 in the figure are parallel.

        Solution: Since angle 1 and angle
                  2 are both 90 degrees, they are
                  supplementary.  By the statement
                  above, they (lines a and b) are
                  parallel.

Youtube video:

http://www.youtube.com/watch?v=RQgJTo52UZw&list=PLECAD1D8F2A3ED403