Download Practice B - MsRLovesMath

Name———————————————————————— Lesson
Lesson 2.4
2.4
Date —————————————
Practice B
For 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 (labeled R).
E
F
m
H
R
n
5. The points E and F lie on a line
6. The planes Q and R intersect in a line
Q
(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?
2-50
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(labeled m).
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Name———————————————————————— Lesson
2.4
Date —————————————
Practice B continued
For use with the lesson “Use Postulates and Diagrams”
12. Points A, B, D, and J are coplanar.
13. ∠ EBA is a right angle.
H
G
Lesson 2.4
In Exercises 12–19, use the diagram to determine if the statement is
true or false.
K
E
14. Points E, G, and A are collinear.
@##$ ⊥ plane H
15.​FG​
A
16. ∠ ABD and ∠ EBC are vertical angles.
##$. 
17. Planes H and K intersect at ​@AB ​
B
C
D
J
F
@##$ and DE​
@​ ##$ intersect.
18.​FG​
19. ∠GCA and ∠CBD are congruent angles.
20. Neighborhood Map A friend e-mailed you the following statements about a
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
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.
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This statement is true because ?3 and 4 are vertical 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 2.4 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.
20. Sample answer: 21. Sample answer:
B
R
P
T
A
S
22. yes; directly indicated by right angle symbol
##$ ⊥ plane S, so it is ⊥ to every line in
23. yes; ​@EF ​
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
##$ intersects @​EF ​
##$ 
27. no; can’t assume that ​@DH​
##$ ⊥ to every line in plane S that it
28. yes; ​@EF ​
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.
2.
l A
A
m
B
C
3.
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
A20
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
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
answers
Lesson 2.3 Apply Deductive
Reasoning, continued
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Lesson 2.4 Use Postulates and
Diagrams, continued
20. a.
1
B
2
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
3. If
then
then
2.
4. If
If
then
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
##$ and ​@DE​
##$ intersect at point E; ​
exactly one point; ​@BE ​
@##$
@##$ intersect at point F.
AF ​ and ​DF​
28. a–b. Sample sketch:
S
then
A
G
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5. Sample answer: Through A and B there exists
##$.  6. Sample answer: The
exactly the one line, ​@AB ​
line @BC​
​ ##$ contains at least the two points B and C.
##$ and ​@AC ​
##$ 
7. Sample answer: The two lines ​@AB ​
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.
13. S
B
A
m
C
A
N
15.
G
D
E
C
B
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
##$. 
the line @​AB ​
Study Guide
1. Postulate 5 2. Postulate 9
3.
C
A
E
B
D
4. true 5. false 6. false
7. true 8. true
9. false
Problem Solving Workshop:
Mixed Problem Solving
T
14. L
F
H
16. yes; Because the line is drawn, you can
assume that A, B, and D are collinear and form
opposite rays. 17. yes; You can assume that the
angles are a linear pair. 18. yes; ∠ BAC is marked
as a right angle and these angles are a linear pair,
so it follows that each has a measure of 908. 19.
no; ∠ BAC is marked as a right angle, but you cannot assume that every line in S through A is a right
angle. 20. yes; From the appearance
@##$ lies in
of the diagram, you can assume that ​BD​
each plane. 21. no; C is clearly in T and not in S,
@##$ cannot lie entirely in plane S. 22. yes; the
so ​CG​
}
congruence marks show that ​BD​ is bisected.
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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answers
A
3
}
23. no; Because BD​
​  is contained in plane T,
it cannot be bisected by T. 24. false; Sample
A21
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