1. Chapter 3 Notes

2. 3.1 Lines and Angles

3. Two lines are PARALLEL if they are COPLANAR and do not INTERSECT
Two lines are SKEW if they are NOT COPLANAR and do not INTERSECT B
Arrows on line mean they are parallel F A E C H D G

4. Two planes that do not Intersect are called PARALLEL planes.
A line and a plane are parallel if they do not intersect. B F A E C H D G

5. Line segments and rays can be parallel too!
As long as the lines going through them are also parallel. O O R R Y Y T T

6. Let’s name some parallel planes, lines, and some skew lines.C B G E H F

7. Parallel line postulate If there is a line and a point not on the line, there is EXACTLY one parallel line through the given point.
Perpendicular line postulate  If there is a line and a point not on the line, there is EXACTLY one perpendicular line through the given point.
Not Parallel

8. A ______________ is a line that INTERSECTS two or more COPLANAR lines at different points.
Two angles are ________________ ANGLES if they occupy ________________ positions. 1 2 3 4
Two angles are ________________ __________________ if they LIE _________ the two lines on __________ sides of the ________________. 5 6 7 8
Two angles are _____________________________ if they LIE __________ the two lines on ___________ sides of the TRANSVERSAL.
Two angles are ______________________________ (also called same side interior angles) if they LIE _________ the two lines on the _______ sides of the TRANSVERSAL.

9. Given a point off a line, draw a line perpendicular to line from given point.
1) From the given point, pick any arc and mark the circle left and right.
2) Those two marks are your endpoints, and construct a perpendicular bisector just like the previous slide.

10. 3.3 – Parallel Lines and Transversals

11. Corresponding Angles Postulate (CAP)
If two lines cut by transversal are ||, then the corresponding angles are congruent m 1 n 2

12. Alternate Interior Angles Theorem (AIA Thrm)1 2 m
If two lines cut by transversal are ||, then the alternate interior angles are congruent 3 4 5 n 6
Consecutive Interior Angles Theorem (CIA Thrm)
Alternate Exterior Angles Theorem (AEA Thrm)
If two lines cut by transversal are ||, then the consecutive interior angles are supplementary
If two lines cut by transversal are ||, then the alternate exterior angles are congruent

13. Perpendicular Transversal
If a transversal is perpendicular to one of two || lines, then it is perpendicular to the other. 1 m 2 n t

14. Find the measure of angles 1 – 7 given the information below.2 3 4 5 6 7
800

15. Find x, y

16. Find x, y, and the measure of all angles3 1 4 2

17. Find w, x, y, z, and the measure of all angles1 4 2 3 5

18. 3.4 – Proving Lines are Parallel

19. Simply stated, the postulates and theorems yesterday have TRUE converses1 n 2 3 p 5

20. m 1 n 4 2 3 p 5

21. I show the angles, you say what theorem makes the lines parallel.
1,5 congruent 1 2 n 4
3,6 congruent 3 5 6 p 7 8
3,5 supplementary
1,8 congruent
4, 8 congruent
5, 8 congruent
3, 5 congruent

22. Which lines are parallel?35 40 38 35 B D

23. You try it! Are l and m parallel? How?

24. Which lines are parallel? How?m p n
40o
80o
50o
80o
discuss

25. You try it! What does x have to be for l and m to be parallel?
(x + 40)o
70o xo
(3x)o

26. m 1 2 n 3 4 5 6 p 7 8

27. 3.5 – Using Properties of Parallel Lines

28. Copy an angle.
1) Draw a ray
2) Use original vertex, make radius.
3) Transfer radius to the ray you drew, and draw an arc.
4) Set radius from D and E, and transfer it to the new lines, setting the point on F and draw an intersection on the arc, then connect the dots.

29. Given a line and a point, construct a line parallel to the given line through the given point.
1) Pick any point on the line, draw a line from there through the given point.
2) Using the angle formed by the given line and the drawn line, make a congruent angle using the given point as the vertex.

30. 3.6 – Parallel Lines in the Coordinate Plane

31. Find points and label Plug into formula Reduce Fraction
SLOPE FORMULA!! MEMORIZE!!
y2 – y1
Find points and label
Plug into formula
Reduce Fraction
(1, 0) (4, -1)
x1 y1 x2 y2
SLOPE = m =
x2 – x1 x y
SLOPE = m =

32. Find points and label Plug into formula Reduce Fraction
SLOPE FORMULA!! MEMORIZE!!
y2 – y1
Find points and label
Plug into formula
Reduce Fraction
(-2, -1) (2, 5)
x1 y1 x2 y2
SLOPE = m =
x2 – x1 x y
SLOPE = m =

33. Postulate: Slopes of Parallel Lines
In a coordinate plane, two nonvertical lines are parallel IFF they have the same slope. Any two vertical lines are parallel.
Basically  Same slope means parallel.
Find the slope between each set of points. See which ones match up to be parallel.
(4, 3)
(-2, -1)
(2, 0)
(-1, 3)
(2, 3)
(-2, -1)
(-5, 2)
(-1, -2)
(-1, 3)
(-3, 0)
(1, 2)
(-8, -4)

34. Slope-intercept form Point-slope form Standard form
Write the equation of the line given a point and a slope in SLOPE-INTERCEPT FORM

36. x y x y

38. Grade of a road, it’s rise over run, then changed into a percent.2
100

39. 3.7 – Perpendicular Lines in the Coordinate Plane

40. Solve for y, change it to ‘y =‘
Distribute
Get y by itself
Notice how by solving for y, we put it in slope intercept form, now we can find the slope.

41. Parallel and Perpendicular Lines
Parallel Lines have the ___________ slope
Green
Blue
What do you notice about the lines and the slope?
Slopes are opposite reciprocals, or slopes multiply to equal -1
Also, vertical and horizontal lines are perpendicular

42. Parallel Lines, SAME SLOPE
Perpendicular Lines, opposite reciprocal.
State the slopes of the line parallel and perpendicular to the slopes on the left.
Slope Parallel Perpendicular

43. Find the slope between each set of points
Find the slope between each set of points. See which ones match up to be perpendicular.
(4, 3)
(-2, -1)
(2, 0)
(-1, 3)
(2, 3)
(-2, -1)
(-5, 3)
(1, -2)
(-1, 3)
(-3, 0)
(3, 2)
(0, 4)

44. Find the slope of each line, then pair up the perpendicular and parallel lines.

46. x y x y