1. By: Esteban Lara Journal #3

2. Parallel Lines and Planes Parallel lines are lines on the same plane that never intersect. And parallel plane are plane that never intersect too. Skew lines are those lines which aren’t on the same plane but don’t intersect either. An example could be the one Mr. Turner uses in class, the roof against the wall and the floor against another wall but on the side of the room. Examples are on the next page.

3. Example Parallel lines in the example to the left would be red 1 and red 2. Parallel planes would be the top red, black, green and blue with the bottom. And skew lines would be red and gray from the right.

4. Transversals A transversal is a line that passes through two other lines (or more than two) intersecting each one. Here are examples. The red line is the transversal.

5. Angles Corresponding: Angles that lie on the same position in comparison to the transversal. Alternate exterior: Lie on opposite sides of the transversal and outside the lines cut by the transversal. Alternate interior: Lie on opposite side of the transversal and are non-adjacent. Same side interior (consecutive interior): Lie on the same side of the transversal between the lines.

6. Examples

7. Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. The converse is: If pairs of corresponding angles are congruent, then the two parallel lines are cut by a transversal. Examples: Those angles are congruent, therefore, the lines are parallel.

8. Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Converse: If pairs of interior angles are congruent, then two parallel lines are cut by a transversal. Examples:

9. Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. The converse is: If two pairs of same-side interior angles are supplementary, then the two line cut by the transversal are parallel.

10. Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. The converse is: If two pairs of alternate exterior angles are congruent, then the two lines cut by the transversal are parallel.

11. Perpendicular Transversal Theorems In a plane, if a transversal is perpendicular to one of two parallel lines, then it’s perpendicular to the other line. Examples:

12. How to find Slope To find the slope of a line you must use the equation m=(y2-y1)/(x2-x1), where y2-y1 is called the rise and x2-x1 is called the run. If you have two lines that have the same slope, then they are PARALLEL. If you have two lines where the product of the slopes of each line equal -1, then they are perpendicular.

13. Examples If the coordinates for a on number two are (1,2) and the coordinate of b are (-1,-2), then you write the equation. (2+2)/(1+1). If the coordinates for a on number 4 are (6, -7) and for b they’re (8, -10) then you write the equation: (-7+10)/(6-8) and you now know that the slope for cd is the same since they’re parallel. If the coordinates in number 6 for a and b are (-5, -5) for a and (-7, -3) for b, then you know the slope will be the same since they’re parallel.

14. Intercept and Point/Slope Form The slope intercept slope is y=mx+b. It’s just an equation to write the coordinates of a line when you already know the slope. You just have to plug in the coordinates for x and y and you get b, then you plug b (which tells you how much you have to go up or down the y axis) and you can complete the line just by knowing the slope and that intersection on the y axis. Point/Slope form is y-y1=m(x-x1). You would use this type of equation when you want to plug in the coordinate of a line already knowing the coordinates of one point and the slope the line should have.

15. Examples Lets say you have a slope of ½ and the coordinates (2,3), plug in the coordinates and slope. 3=1/2(2)+b. Now you just solve for b. After you solve for b, you just have to put a point where b tells you and with the slope you continue graphing the line. Lets say you have the points (1, 2) and a slope of 2/3. You plug in 1 on x1 and 1 on y1, and of course the slope on m. Now you know how much you have to move the point along the x and y and the slope will lead you to draw the rest. Now imagine that Mr. Turner tells us to draw a line with the coordinates (5, 10) and a slope of ½. What we have to do is replace x1 and y1 with the 5 and 10 and place the slope in the formula. And draw the point and a line based on the slope.

16. Transitive Property in Parallel and Perpendicular lines Transitive property can apply to parallel lines in the sense that when a line is parallel to another, the slope of a can be the same as the slope of b and since those lines are parallel, then c must also have the same slope as a and b. When the lines are perpendicular we can say that the first angle created by the two lines is the same degree angle as the second and third angles when we prove that it’s the same as the fourth angle (or even if you prove that an angle is the same as another one and the lines are perpendicular, then the rest of the angles are also congruent.

17. Examples We know the slope of 1 is 3, since it’s parallel to line 2, then we know the slope will also be 3. We know the measurement of a is congruent to b and b is a right angle, so the measurement of d and c must also be 90 degrees. If the red line crosses the black line perpendicularly, and b measures 90 degrees, then a, d and c are also right angles.

18. Perpendicular Line Theorems Perpendicular line theorems tell you how you can prove that two lines are parallel by just knowing if they are perpendicular or not. You can also tell what the measurements of the angles will be if one line is crossed by another perpendicular to that one. In a theorem it tells you if the lines are perpendicular or not depending on measurement of the angles. Examples: If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. If two coplanar lines are perpendicular to the same line, then the lines are parallel to each other.

19. Biography http://cosketch.com/Rooms/nbpgynb http://www.geom.uiuc.edu/~dwiggins/pict16.GIF