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Lesson 2.2Betweenness
pp. 45-47
Objectives:1. To define betweenness of points.2. To define more subsets of lines.3. To apply correct notation to the
new geometric terms.
Objectives:1. To define betweenness of points.2. To define more subsets of lines.3. To apply correct notation to the
new geometric terms.
DefinitionDefinitionDefinitionDefinition
B is between A and C if
BC BA = {B} when A, B, and
C are collinear. In symbols,
you can write A-B-C.
B is between A and C if
BC BA = {B} when A, B, and
C are collinear. In symbols,
you can write A-B-C.
AA BB CC
In order for B to between A and C, all three points must
be collinear!
In order for B to between A and C, all three points must
be collinear!
EXAMPLE 1: Show that C is not between A and B. EXAMPLE 1: Show that C is not between A and B.
If C is between A and B, then by
the definition of betweenness,
CA CB = {C}, which is false.
Notice that CA CB = CA.
If C is between A and B, then by
the definition of betweenness,
CA CB = {C}, which is false.
Notice that CA CB = CA.
AA BB CC
BA and BC are opposite rays if
B is between A and C.
BA and BC are opposite rays if
B is between A and C.
DefinitionDefinitionDefinitionDefinition
A segment is the set consisting of two points A and B and all the points in between. The symbol for segment AB is AB. AB = {A,B} {x|A-X-B}.
A segment is the set consisting of two points A and B and all the points in between. The symbol for segment AB is AB. AB = {A,B} {x|A-X-B}.
DefinitionDefinitionDefinitionDefinition
Symbol MeaningSymbol Meaning
A point A
AB line AB
XY half-line XY
CD ray CD
AC segment AC
LM vector LM
A point A
AB line AB
XY half-line XY
CD ray CD
AC segment AC
LM vector LM
AA BB CC DD
Example:
AB CB =
Example:
AB CB =
1. {B}
2. AC
3. AC
4. AD
5. Both 3 & 4
1. {B}
2. AC
3. AC
4. AD
5. Both 3 & 4
Example:
AB CB =
Example:
AB CB =
1. {B}
2. AC
3. AC
4. AD
5. Both 1 & 2
1. {B}
2. AC
3. AC
4. AD
5. Both 1 & 2
AA BB CC DD
Example:
AB CD =
Example:
AB CD =
1. { }
2. AD
3. CD
4. CD
1. { }
2. AD
3. CD
4. CD
AA BB CC DD
A B C
E
D
1. Is A between E and B?1. Yes 2. No
1. Is A between E and B?1. Yes 2. No
Example: Example:
A B C
E
D
2. Identify a correct betweenness statement1. A-B-C 2. E-D-C
2. Identify a correct betweenness statement1. A-B-C 2. E-D-C
Example: Example:
A B C
E
D
3. Name four segments.3. Name four segments.
Example: Example:
A B C
E
D
4. Are CA and AC opposite rays?1. Yes 2. No
4. Are CA and AC opposite rays?1. Yes 2. No
Example: Example:
A B C
E
D
5. Name two pairs of opposite rays.
5. Name two pairs of opposite rays.
Example: Example:
Homeworkpp. 46-47
Homeworkpp. 46-47
►A. ExercisesUse the figure for exercises 6-12.►A. ExercisesUse the figure for exercises 6-12.
9. Name two pairs of opposite rays.9. Name two pairs of opposite rays.
FF
BB
LL
KK
MM
CC
►A. ExercisesUse the figure for exercises 6-12.►A. ExercisesUse the figure for exercises 6-12.
11. Name four line segments.11. Name four line segments.
FF
BB
LL
KK
MM
CC
AA BB CC DD
13. AB BC13. AB BC
►B. ExercisesUse the figure below and correct notation to describe the set of points in exercises 13-21.
►B. ExercisesUse the figure below and correct notation to describe the set of points in exercises 13-21.
15. BC AC15. BC AC
►B. ExercisesUse the figure below and correct notation to describe the set of points in exercises 13-21.
►B. ExercisesUse the figure below and correct notation to describe the set of points in exercises 13-21.
AA BB CC DD
►B. ExercisesUse the figure below and correct notation to describe the set of points in exercises 13-21.
►B. ExercisesUse the figure below and correct notation to describe the set of points in exercises 13-21.
17. {A} AB17. {A} AB
AA BB CC DD
19. DC AC19. DC AC
►B. ExercisesUse the figure below and correct notation to describe the set of points in exercises 13-21.
►B. ExercisesUse the figure below and correct notation to describe the set of points in exercises 13-21.
AA BB CC DD
AA BB CC DD
►B. Exercises21. Name two rays that have B as their
endpoint. What is the special name that these two rays have?
►B. Exercises21. Name two rays that have B as their
endpoint. What is the special name that these two rays have?
►B. Exercises►B. Exercises23. What is the intersection of AX and
XA?23. What is the intersection of AX and
XA?
►C. Exercises►C. Exercises25. What is the intersection of AB and
BA?25. What is the intersection of AB and
BA?
■ Cumulative ReviewName the postulate that justifies each statement.26. B is between A and C; therefore BA
{B} BC = AC
■ Cumulative ReviewName the postulate that justifies each statement.26. B is between A and C; therefore BA
{B} BC = AC
■ Cumulative ReviewName the postulate that justifies each statement.27. A, B, and C are collinear. B, C, and D
are also collinear. Therefore A, C, and
D are collinear, and AB = BC = CD.
■ Cumulative ReviewName the postulate that justifies each statement.27. A, B, and C are collinear. B, C, and D
are also collinear. Therefore A, C, and
D are collinear, and AB = BC = CD.
■ Cumulative ReviewAssume that A-B-C and X-B-Y. What kind of lines are AC and XY, if 28. A, X, and B are not collinear.
■ Cumulative ReviewAssume that A-B-C and X-B-Y. What kind of lines are AC and XY, if 28. A, X, and B are not collinear.
■ Cumulative ReviewAssume that A-B-C and X-B-Y. What kind of lines are AC and XY, if 29. A, X, and B are collinear.
■ Cumulative ReviewAssume that A-B-C and X-B-Y. What kind of lines are AC and XY, if 29. A, X, and B are collinear.
■ Cumulative ReviewAssume that A-B-C and X-B-Y. What kind of lines are AC and XY, if 30. Suppose B is between A and C and C
is between B and D. Draw two conclusions (include a sketch).
■ Cumulative ReviewAssume that A-B-C and X-B-Y. What kind of lines are AC and XY, if 30. Suppose B is between A and C and C
is between B and D. Draw two conclusions (include a sketch).