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Practice BFor use with the lesson “Use Postulates and Diagrams”

Draw a sketch to illustrate each postulate.

1. If two lines intersect, then their intersection is exactly one point.

2. If two points lie in a plane, then the line containing them lies in the plane.

3. If two planes intersect, then their intersection is a line.

Use the diagram to state and write out the postulate that verifies the truth of the statement.

4. The points E, F, and H lie in a plane

Q

Rn

mE

FH

(labeled R).

5. The points E and F lie on a line (labeled m).

6. The planes Q and R intersect in a line (labeled n).

7. The points E and F lie in a plane R. Therefore, line m lies in plane R.

In Exercises 8–11, think of the intersection of the ceiling and the front wall of your classroom as line k. Think of the center of the floor as point A and the center of the ceiling as point B.

8. Is there more than one line that contains both points A and B?

9. Is there more than one plane that contains both points A and B?

10. Is there a plane that contains line k and point A?

11. Is there a plane that contains points A, B, and a point on the front wall?

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In Exercises 12–19, use the diagram to determine if the statement is true or false.

12. Points A, B, D, and J are coplanar. KH

A B C

G

F

E

D

J

13. ∠ EBA is a right angle.

14. Points E, G, and A are collinear.

15. @##$ FG ⊥ plane H

16. ∠ ABD and ∠ EBC are vertical angles.

17. Planes H and K intersect at @##$ AB .

18. @##$ FG and @##$ DE intersect.

19. ∠GCA and ∠CBD are congruent angles.

20. Neighborhood Map A friend e-mailed you the following statements about a neighborhood. Use the statements to complete parts (a)–(e).

Building B is due south of Building A.

Buildings A and B are on Street 1.

Building C is due east of Building B.

Buildings B and C are on Street 2.

Building D is southeast of Building B.

Buildings B and D are on Street 3.

Building E is due west of Building C.

∠ DBE formed by Streets 2 and 3 is acute.

a. Draw a diagram of the neighborhood.

b. Where do Streets 1 and 2 intersect?

c. Classify the angle formed by Streets 1 and 2.

d. What street is building E on?

e. Is building E between buildings B and C? Explain.

Practice B continuedFor use with the lesson “Use Postulates and Diagrams”

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Lesson Apply Deductive Reasoning, continued This statement is true because ?3 and 4 are verti-cal angles and therefore their measures are equal. If m∠ 3 5 658, then m∠ 5 5 658. This statement is true because you are given ∠ 3 > ∠ 5. If m∠ 5 5 658, then m∠ 6 5 1158. This statement is true because ?5 and 6 are supplementary angles and therefore their measures add up to 1808. By the Law of Syllogism, you can make the following conclusion. If m∠ 2 5 1158, then m∠ 6 5 1158.

Lesson Use Postulates and Diagrams

Teaching Guide

1. dot; line with two arrowheads (but it extends without end); a floor, or a wall (but it extends without end)

2. Answers will vary. (Points should lie on the same line.); Answers will vary. (Points should lie in the same plane.) 3. A postulate is a rule that is accepted without proof (axiom). A theorem is a rule that can be proved. 4. 1808; a straight line 5. A linear pair is formed by two adjacent angles whose noncommon sides are opposite rays. Vertical angles are formed by two pairs of opposite rays.

Investigating Geometry Activity

1. one 2. one 3. one; three 4. lies in the plane

5. line

Practice Level A

1. Postulate 5 2. Postulate 8 3. Postulate 9

4. Postulate 6 5. Postulate 6 6. Postulate 10

7. Postulate 8 8. Postulate 9 9. Through the two points A and B, there exists exactly the one line, q. 10. Line q contains at least the two points A and B. 11. Lines p and q intersect in exactly the one point A. 12. Through the three noncollinear points C, D, and E, there exists only the one plane S. 13. Plane S contains at least the three noncollinear points C, D, and E.

14. Sample answer: The two points D and E lie in plane R, so the line m that contains them lies in R.

15. The intersection of planes R and S is line m.

16. no 17. no 18. yes 19. no

20. Sample answer: 21. Sample answer:

AP

B

R

S

T

22. yes; directly indicated by right angle symbol

23. yes; @##$ EF ⊥ plane S, so it is ⊥ to every line in S that it intersects 24. no; can’t assume collinearity without the line drawn

25. yes; all 3 points are on one line

26. yes; it is obvious in diagram

27. no; can’t assume that @##$ DH intersects @##$ EF

28. yes; @##$ EF ⊥ to every line in plane S that it intersects 29. no; can’t assume these lines ⊥ without a rt. angle marked

30. a. The pole is perpendicular to the ground. b. yes c. The corresponding segments are marked >, so the distances are all 6 ft. d. yes; because the pole is perpendicular to the ground, it is perpendicular to each line passing through point P.

Practice Level B

1–3. Sample sketches are given. 1.

mA

l 2. A

C

B

3.

4. Postulate 8: Through any three noncollinear points there exists exactly one plane.

5. Postulate 5: Through any two points there exists exactly one line. 6. Postulate 11: If two planes intersect, then their intersection is a line.

7. Postulate 10: If two points lie in a plane, then the line containing them lies in the plane.

8. No. Through any two points there exists exactly one line. 9. Yes. Points A and B could lie on the line intersecting two planes.

10. Yes. Take point A and any two points on line k and you can form a plane through those three points that contains all of line k. 11. Yes. The plane that runs from the front of the room to the back of the room through points A and B contains both points and a point on the front wall.

12. true 13. false 14. false 15. false 16. true 17. true 18. false 19. false

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Lesson Use Postulates and Diagrams, continued 20. a.

3

1

2

A

B E C

D

b. building B c. right d. 2 e. Yes, because ∠ DBE is acute and Building E is due west of Building C.

Practice Level C

1. If then 2. If then

3. If then 4. If then

5. Sample answer: Through A and B there exists exactly the one line, @##$ AB . 6. Sample answer: The line @##$ BC contains at least the two points B and C.

7. Sample answer: The two lines @##$ AB and @##$ AC intersect in exactly the one point, A.

8. Through the three noncollinear points A, B, and C there exists exactly the one plane, T. 9. The plane T contains at least the three points A, B, and C. 10. Sample answer: The points A and C lie in plane T and so does the line @##$ AC .

11. The planes U and T intersect in the line @##$ BC .

Sample answers for Exercises 12–15:

12. A C

B 13. S

T

m

14. L

NA

15. D

E F

H

G

16. yes; Because the line is drawn, you can assume that A, B, and D are collinear and form opposite rays. 17. yes; You can assume that the angles are a linear pair. 18. yes; ∠ BAC is marked as a right angle and these angles are a linear pair, so it follows that each has a measure of 908. 19. no; ∠ BAC is marked as a right angle, but you can-not assume that every line in S through A is a right angle. 20. yes; From the appearance of the diagram, you can assume that @##$ BD lies in each plane. 21. no; C is clearly in T and not in S, so @##$ CG cannot lie entirely in plane S. 22. yes; the congruence marks show that } BD is bisected.

23. no; Because } BD is contained in plane T, it cannot be bisected by T. 24. false; Sample answer: If three hinges on a door are collinear points, then different positions of the door represent different planes through those points.

25. false; Sample answer: A single plane cannot be passed through the vertices of a triangular pyramid. 26. true

27. If two lines intersect, then their intersection is exactly one point; @##$ BE and @##$ DE intersect at point E; @##$ AF and @##$ DF intersect at point F.

28. a–b. Sample sketch: S

A CBG

c. Postulate 8: Through the three noncollinear points A, B, and C, there exists the one plane, G. Postulate 11: The two planes S and G intersect in the line @##$ AB .

Study Guide

1. Postulate 5 2. Postulate 9

3. C

A E B

D

4. true 5. false 6. false

7. true 8. true 9. false

Problem Solving Workshop: Mixed Problem Solving

1. a. The amount of bacteria doubles after every hour. b. 768 billion bacteria

2. a. Inductive reasoning; it is based on a pattern in the data. b. Deductive reasoning; you are using values that are given on the graph. c. Inductive reasoning; it is based on a pattern in the data.

3. Answers will vary. 4. a. may not; You can write the statement as “If Adam lives at Pine Meadows, then he is not allowed to have a dog.” b. may have; All you know is that Jodi was in Utah and the Zion National Park is in Utah. You do not know if she visited it. 5. 43

6. a. true b. false; An earthquake is also felt if its Richter magnitude is 4.

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