2. Points, Lines, Slopes– applications

3. EXAMPLE 1 Identify relationships in space d. Plane ( s ) parallel to plane EFG and containing point A c. Line ( s ) perpendicular to CD and containing point A a. Line ( s ) parallel to CD and containing point A b. Line ( s ) skew to CD and containing point A Think of each segment in the figure as part of a line. Which line ( s ) or plane(s) in the figure appear to fit the description?

4. EXAMPLE 1 SOLUTION Identify relationships in space AB, HG, and EF all appear parallel to CD, but only AB contains point A. a. Both AG and AH appear skew to CD and contain point A. b. d. Plane ABC appears parallel to plane EFG and contains point A. c. BC, AD, DE, and FC all appear perpendicular to CD, but only AD contains point A.

5. EXAMPLE 2 Identify parallel and perpendicular lines Name a pair of parallel lines. a. Name a pair of perpendicular lines. b. Is FE AC ? Explain. c. Photography The given line markings show how the roads are related to one another.

6. EXAMPLE 2 b. MD BF Identify parallel and perpendicular lines FE is not parallel to AC, because MD is parallel to FE and by the Parallel Postulate there is exactly one line parallel to FE through M. c. SOLUTION a. MD FE

7. Ways to Prove Two Lines Parallel Ways to Prove Two Lines Parallel show that a pair of corresponding angles are congruent show that a pair of corresponding angles are congruent show that a pair of alternate interior angles are congruent show that a pair of alternate interior angles are congruent show that a pair of same-side interior are supplementary show that a pair of same-side interior are supplementary in a plane show that both lines are perpendicular to a 3 rd line in a plane show that both lines are perpendicular to a 3 rd line show that both lines are parallel to a 3 rd line show that both lines are parallel to a 3 rd line

8. Quick Review What is the difference between collinear points and coplanar lines? What happens when two planes intersect? What are the 4 ways you can create a plane? – Answer: collinear points are points on the same line and coplanar lines are lines in the same plane - Answer: ONE line is formed -Answer: THREE noncollinear points, a line and a point off the line, two intersecting lines, two parallel lines

9. Geometry Introductory Project - Spatial Reasoning

10. 1. Two lines are coplanar - sometimes The crosswalk on the way to Maria's house shows two lines that are on the same plane. Lines l and n are both in plane X. X l n

11. 1. Two lines are coplanar - sometimes In this picture (where Anna enjoys a lovely invisible steak), lines k and f are non-coplanar, or not on the same plane. k f

12. 2. A line intersects a plane in one point - sometimes Sewing is hard, so we looked up a few stitches online and this is what came up! Line n (needle) intersects plane Y (fabric) in one single point.. Y n

13. 2. A line intersects a plane in one point - sometimes Line r is parallel to plane G, so they don't intersect with each other. G r

14. 3. Two planes intersect in a line - sometimes This model airplane is a good example of two planes, W and B, intersecting in line m. W B m

15. 3. Two planes intersect in a line - sometimes Geometry students can play with kid's toys, too! Planes M and A are parallel, so they will never intersect. M A

16. 4. Two planes intersect in a single point - never Two planes never intersect in a single point because planes go on forever and do not have an edge. If two planes do intersect, then their intersection is a line.

17. 5. Planes have an edge - never Planes do not have an edge because they go on forever.

18. 6. Two points are collinear - always This car definitely caught our eye, with its wheels like two points, b and f, that can be connected by a line (c ). Well, and the gold and silver duct tape... b f c

19. 7. Three points are collinear - sometimes As we were waiting for the bus, we saw that these three holes on the bus stop sign, points a, m, and l, are all on line k. They are collinear.... a m l k

20. 7. Three points are collinear - sometimes In this example, points o and m are collinear, but point g is not. Any two of them could be collinear, but not the third.... o m g l

21. 8. Three points are coplanar - always As Maria was emptying her coin purse, she found that 3 pennies happened to be on the same plane. p, a, and l are coplanar. p a l

22. 9. Four points are coplanar - sometimes During a fun game of parcheesi, we noticed that the four dots on a dice, u, d, t, and c, are on the same plane.... c u d t.

23. 9. Four points are coplanar - sometimes These four rocks, points k, p, g, and r, are not coplanar, or on one single plane. k p g r

24. 10. A point is a small, filled circle - never A point is represented by a small filled circle, but it is just a location in space. A point has no dimension.

26. The airplanes are in different horizontal planes

27. a.line b.b. point

97. 9 10 11 12 13 14 15 16

100. 3 and 4

135. FIND SLOPE

136. Using slope as a rate of change

153. STEP 1

154. Step 2

156. STEP 4