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Postulate Problems This is for high school students.
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Millau BridgeSir Norman Foster
Point, Lines, Planes, Angles
FallingwatersFrank Lloyd Wright
Millenium ParkFrank Lloyd Wright
1.5 Written Exercises Page 25
1.5 Written Exercises Page 25
1. State Theorem 1-2 using the phrase “one and only one.”
Theorem 1-2Through a line and a point not in the line there is exactly one plane.
Through a line and a point not in the line there one and only one plane.
2. Reword Theorem 1-3 as two statements, one describing existence and the other describing uniqueness.
Theorem 1-3If two lines intersect, there is exactly one line that contains the lines.
Existence: If 2 lines intersect, there is at least one plane that contains the lines.
Uniqueness: If 2 lines intersect, there is no more than one plane that contains the lines.
3. Planes M and N are known to intersect.
a] What kind of figure is the intersection.
b] State the postulate that supports your answer.
4. Points A and B are known to lie in a plane.
a] What can you say about .AB�������������� �
b] State the postulate that supports your answer.
A line
3. Planes M and N are known to intersect.
a] What kind of figure is the intersection.
b] State the postulate that supports your answer.
4. Points A and B are known to lie in a plane.
a] What can you say about .AB�������������� �
b] State the postulate that supports your answer.
A line
F
ED
CB
A
If 2 planes intersect, their intersection is a line.
.ABlies in the plane�������������� �
If 2 points are in a plane, the line that contains them is in the plane.
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
5. Write the postulate that assures you that exists.
6. Name a plane that contains
HG
FE
D C
B
A
AC�������������� �
AC�������������� �
HG
FE
D C
B
A
Two points contain exactly 1 line. or
Two points determine a line.
ABCD
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
7. Name a plane the contains but is not shown.
AC�������������� �
HG
FE
D C
B
A
HG
FE
D C
B
A
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
7. Name a plane the contains but is not shown.
8. Name the intersection of plane DCFE and plane ABCD.
AC�������������� �
ACGE
HG
FE
D C
B
A
HG
FE
D C
B
A
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
7. Name a plane the contains but is not shown.
8. Name the intersection of plane DCFE and plane ABCD.
AC�������������� �
DC�������������� �
ACGE
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
9. Name 4 lines in the diagram that do not intersect. AC
�������������� �H
G
FE
D C
B
A
HG
FE
D C
B
A
, , ,AD BC EH FG�������������������������������������������������������������������������������������������� �������
, , ,AE BF DH CG�������������������������������������������������������������������������������������������� �������or
or
, , ,AB EF DC HG�������������������������������������������������������������������������������������������� �������
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
10. Name 2 lines that are not shown in the diagram and that do not intersect.
HG
FE
D C
B
A
HG
FE
D C
B
A
,AC EG���������������������������������������� ���
or
or,ED FC���������������������������������������� ���
,DB EG���������������������������������������� ���
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
11. Name 3 planes that do not intersect and do not contain
EF�������������� �
HG
FE
D C
B
A
EF�������������� �
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
11. Name 3 planes that do not intersect and do not contain
EF�������������� �
HG
FE
D C
B
A
EF�������������� �
HG
FE
D C
B
A
ABCD, DCGH, ???
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
11. Name 3 planes that do not intersect and do not contain
EF�������������� �
HG
FE
D C
B
A
EF�������������� �
HG
FE
D C
B
A
ABCD, DCGH, ???
HG
FE
D C
B
A
ABGH
In exercises 5-11 you will have to visualize certain lines and planes not shown in the diagram of the box. When you name a plane, name it using 4 points, no three being collinear.
12. If you measure
HG
FE
D C
B
A
with a protractor you get more than 900. But you know that the angle is 900. Using this example, complete the chart.
In the diagram
obtuse
In the box
right
EFG AEF DCB FBCright
rightrightright
acute obtuse
State whether it is possible for the figure described to exist.
Write yes or no.
13. Two points both lie in each of two lines.
No, it violates the postulate that 2 points are in exactly one line.
State whether it is possible for the figure described to exist.
Write yes or no.
14. Three points all lie in each of two planes.
Yes, why?
State whether it is possible for the figure described to exist.
Write yes or no.
14. Three points all lie in each of two planes.
F
ED
CB
A
Yes, why?
State whether it is possible for the figure described to exist.
Write yes or no.
15. Three noncollinear points all lie in each of 2 planes.
No. It violates the postulate thatThree noncolllinear points lie in exactly one plane.
State whether it is possible for the figure described to exist.
Write yes or no.
16. Two points lie in a plane X, two other points lie in a different plane Y,and the four point are coplanar but not collinear.
Yes.
Show why?
State whether it is possible for the figure described to exist.
Write yes or no.
16. Two points lie in a plane X, two other points lie in a different plane Y,and the four point are coplanar but not collinear.
Yes.
Show why?
State whether it is possible for the figure described to exist.
Write yes or no.
16. Two points lie in a plane X, two other points lie in a different plane Y,and the four point are coplanar but not collinear.
H
GF
E
D
CB
A
Yes. Show why?
A & B are in ABFE
H & G are in DCHG
Yet all points are coplanar.
17. Points R, S, and T are noncollinear points.
a] State the postulate that guarantees the existence of plane X that contains R, S, and T.
b] Draw a diagram showing the above.
Three noncollinear points are contained in at least one plane.
Through three noncollinear points there is exactly on plane that contains them.
or
17. Points R, S, and T are noncollinear points.
a] State the postulate that guarantees the existence of plane X that contains R, S, and T.
b] Draw a diagram showing the above.
Three noncollinear points are contained in at least one plane.
Through three noncollinear points there is exactly on plane that contains them.
or
XD E
F
XD E
F
17. Points R, S, and T are noncollinear points.
c] Suppose that P is any point of other than R or S.
d] State the postulate that guarantees that exists.
Yes. If two points of a line lie in a plane, then the line lies in the plane.
Does P lie in plane X?
RS�������������� �
TR�������������� �
Through any 2 points there exists exactly one plane the contains them.
e] State the postulate that guarantees that is in plane X..
TR�������������� �
If two points of a line lie in a plane, then the line lies in the plane.
BD
C
A
P
18. Points A, B, C, and D are four are noncoplanar points.
a] State the postulate that guarantees the existence of planes ABC, ABD, ACD and BCD.
b] Explain how the Ruler Postulate guarantees the existence of a point P between A and D
If two points of a line lie in a plane, then the line lies in the plane.
Through any 2 points there exists exactly one plane the contains them.
BD
C
A
P
18. Points A, B, C, and D are four are noncoplanar points.
d] Explain why there are an infinite number of planes through
Through any 3 noncollinear points there exists exactly one plane the contains them.
c] State the postulate that guarantees the existence of plane BCP.
BC�������������� �
There are an infinite number of points on Therefore, there are an infinite number of noncollinear points to match up with B and C. Each case forms a unique plane.
AD�������������� �
C’est fini.
Good day and good luck.
