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Lesson 2.2 Betweenness pp. 45-47

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Lesson 2.2 Betweenness pp. 45-47. Objectives: 1.To define betweenness of points. 2.To define more subsets of lines. 3.To apply correct notation to thenew geometric terms. 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. - PowerPoint PPT Presentation
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Lesson 2.2 Betweenness pp. 45-47
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Page 1: Lesson 2.2 Betweenness pp. 45-47

Lesson 2.2Betweenness

pp. 45-47

Page 2: Lesson 2.2 Betweenness 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.

Page 3: Lesson 2.2 Betweenness pp. 45-47

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.

Page 4: Lesson 2.2 Betweenness pp. 45-47

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!

Page 5: Lesson 2.2 Betweenness pp. 45-47

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

Page 6: Lesson 2.2 Betweenness pp. 45-47

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

Page 7: Lesson 2.2 Betweenness pp. 45-47

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

Page 8: Lesson 2.2 Betweenness pp. 45-47

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

Page 9: Lesson 2.2 Betweenness pp. 45-47

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

Page 10: Lesson 2.2 Betweenness pp. 45-47

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

Page 11: Lesson 2.2 Betweenness pp. 45-47

Example:

AB CD =

Example:

AB CD =

1. { }

2. AD

3. CD

4. CD

1. { }

2. AD

3. CD

4. CD

AA BB CC DD

Page 12: Lesson 2.2 Betweenness pp. 45-47

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:

Page 13: Lesson 2.2 Betweenness pp. 45-47

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:

Page 14: Lesson 2.2 Betweenness pp. 45-47

A B C

E

D

3. Name four segments.3. Name four segments.

Example: Example:

Page 15: Lesson 2.2 Betweenness pp. 45-47

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:

Page 16: Lesson 2.2 Betweenness pp. 45-47

A B C

E

D

5. Name two pairs of opposite rays.

5. Name two pairs of opposite rays.

Example: Example:

Page 17: Lesson 2.2 Betweenness pp. 45-47

Homeworkpp. 46-47

Homeworkpp. 46-47

Page 18: Lesson 2.2 Betweenness pp. 45-47

►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

Page 19: Lesson 2.2 Betweenness pp. 45-47

►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

Page 20: Lesson 2.2 Betweenness pp. 45-47

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.

Page 21: Lesson 2.2 Betweenness pp. 45-47

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

Page 22: Lesson 2.2 Betweenness pp. 45-47

►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

Page 23: Lesson 2.2 Betweenness pp. 45-47

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

Page 24: Lesson 2.2 Betweenness pp. 45-47

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?

Page 25: Lesson 2.2 Betweenness pp. 45-47

►B. Exercises►B. Exercises23. What is the intersection of AX and

XA?23. What is the intersection of AX and

XA?

Page 26: Lesson 2.2 Betweenness pp. 45-47

►C. Exercises►C. Exercises25. What is the intersection of AB and

BA?25. What is the intersection of AB and

BA?

Page 27: Lesson 2.2 Betweenness pp. 45-47

■ 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

Page 28: Lesson 2.2 Betweenness pp. 45-47

■ 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.

Page 29: Lesson 2.2 Betweenness pp. 45-47

■ 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.

Page 30: Lesson 2.2 Betweenness pp. 45-47

■ 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.

Page 31: Lesson 2.2 Betweenness pp. 45-47

■ 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).


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