1. Point, Lines, Planes, Angles
Millau Bridge
Sir Norman Foster
Millenium Park
Frank Lloyd Wright
Fallingwaters
Frank Lloyd Wright
Point, Lines, Planes, Angles
1.5 CE Page 24

2. 1.5 CE Page 24 2. State postulate 6 using the phrase one and only one.
Theorem 1.1 states that two lines intersect in exactly one point. The diagram suggests what would happen if you tried to show that 2 lines were drawn through 2 points.
State the postulate that makes this situation impossible.
Through any 2 points there is exactly one line.
2. State postulate 6 using the phrase one and only one.
Post. 6 Through any two points, there is exactly one line.
Through any two points, there is one and only one line.

3. A segment has exactly one midpoint.
3. Reword the following statement as two statements, one describing existence and one describing uniqueness.
A segment has exactly one midpoint.
Existence: All segments have a midpoint.
Uniqueness: A segment has only 1 midpoint.
Postulate 6 is sometimes stated as “Two points determine a line.”
4. Restate Theorem 1-2 using the word determine.
Theorem 1-2
If 2 lines intersect, then they intersect in exactly one point.
If 2 intersecting lines determine one point.

4. 5. Do 2 intersecting lines determine a plane?
Postulate 6 is sometimes stated as “Two points determine a line.”
5. Do 2 intersecting lines determine a plane?
Yes, there are 3 non-collinear points. Also…
Theorem 1-3 states that two intersecting lines are in exactly 1 plane.
6. Do three points determine a line?
Yes. If the points are collinear then they determine 1 line.
If the points are non-collinear, then they determine 3 lines.

5. 7. Do three points determine a plane?
No. If the points are collinear, they don’t. If the points are non-collinear, they do.
Sometimes is always answered NO.

6. State a postulate, or part of a postulate, that justifies your answer for each exercise.
8. Name 2 points that determine line l.
A, C
9. Name three points that determine plane M.
A, B, C also
B, C, D
A, B, D also
10. Name the intersection of planes M and N.

7. Yes, they are just not drawn or labeled.
State a postulate, or part of a postulate, that justifies your answer for each exercise.
11. Does lie in plane M?
Yes, it is just not drawn.
12. Does plane N contain
any points not on ?
Yes, they are just not drawn or labeled.
Each plane has an infinite number of points.

8. The 4 legs may not be in the same plane.
13. Why does a three-legged support work better than one with four legs?
Three legs are always in a plane and therefore are steady and do not rock.
Four points may or may not be in a plane. Also most ground is not level.
14. Explain why a four-legged table may rock even if the floor is level.
The 4 legs may not be in the same plane.
Also over time the four legs move, swell, and are jarred out of position.

9. 15. A carpenter checks to see if a board is warped by laying a straightedge across the board in several directions. State the postulate that is related to this procedure.
If there is space under the straight edge, then the board is curved or warped.
If two points of a line are in the plane, then the line is in the plane.

10. Yes. Note that this situation has 3 non-collinear points.
16. Think of the intersection of of the ceiling and the front wall of your classroom. Let the point in the center of the floor be point C.
a. Is there a plane that contains line L and point C?
Yes. Note that this situation has 3 non-collinear points. L L
b. State the theorem that applies.
Through a line and a point not on the line, there is exactly one plane.

11. C’est fini.
Good day and good luck.