Download Answers for the lesson “Prove Theorems about Perpendicular Lines”

LESSON
3.6
Answers for the lesson “Prove Theorems
about Perpendicular Lines”
Skill Practice
1. }
AB; it’s > to the parallel lines.
2. If two lines intersect to form
a linear pair of congruent angles,
then the lines are perpendicular.
3. If two sides of two adjacent acute
angles are perpendicular, then the
angles are complementary.
4. If two lines are perpendicular,
then they intersect to form four
right angles.
5. 258
6. 908
7. 528
8. Lines Perpendicular to a
Transversal Theorem
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9. Since the two angles labeled x8
form a linear pair of congruent
angles, t >n; since the two lines
are perpendicular to the same line,
they are parallel to each other.
10. Alternate Exterior Angles
Converse Theorem
11. Sample answer: Draw a line.
Construct a second line
perpendicular to the first line.
Construct a third line
perpendicular to the second line.
12. Fold the paper into thirds
lengthwise and then in half across
its width.
13. There is no information to
indicate that y i z or y >x.
‹]›
14. }
AC is not > to AB
.
15. 13
16. 95
17. 33
18. Lines n and p; they are
perpendicular to line k.
19. Lines f and g; they are
perpendicular to line d.
20. Lines z and y; they are
perpendicular to line w. Lines
v, w, and x; lines v and w are
perpendicular to line y, and lines
w and x are perpendicular to
line z.
21. A
22. The distance between nonparallel
lines is not constant.
23. 4.1
24. 2.8
}
}
25. Ï 13 < 3.6; Ï 26 < 5.1
26. Sample answer: mŽ1 5 908 since
it is a right angle; mŽ2 1 308 5
908, therefore mŽ2 5 608;
mŽ5 5 908 since it is vertical
to Ž1; mŽ3 5 308 since it is
vertical to the angle measuring
308; mŽ4 5 208 since
mŽ3 1 mŽ4 1 408 5 908.
27. 2.5
Geometry
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85
28. Construct a perpendicular line
to the plane passing through the
point. Find the length of the
perpendicular segment from the
point to the point where the
segment intersects the plane; if
and only if the line is parallel to
the plane.
Problem Solving
29. Point C; the shortest distance is
the length of the perpendicular
segment.
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30. Check students sketches;
perpendicular segments; shortest
distance between 2 parallel lines
is a perpendicular line, so the
perpendicular line would require
less paint than the diagonal ones.
31. 2. Linear Pair Postulate
3. mŽ1 1 mŽ2 5 1808
5. Definition of congruent angles
8. Division Property of Equality
9. Ž1 is a right angle.
10. Definition of perpendicular
lines
32. Given: a > b
Prove: Ž1, Ž2, Ž3, and Ž4 are
right angles.
Statements (Reasons)
1. a > b
(Given)
2. Ž1 is a right angle. (Definition
of perpendicular lines)
3. mŽ1 5 908
(Definition of
right angle)
4. Ž1 and Ž4 are vertical angles.
(Definition of vertical angles)
5. mŽ4 5 908 (Vertical Angles
Congruence Theorem)
6. Ž4 is a right angle. (Definition
of right angle)
7. Ž1 and Ž2 form a linear pair.
(Definition of a linear pair)
8. mŽ1 1 mŽ2 5 1808
(Linear Pair Postulate)
9. 908 1 mŽ2 5 1808
(Substitution)
10. mŽ2 5 908
(Subtraction
Property of Equality)
11. Ž2 is a right angle.
(Definition of right angle)
12. Ž2 and Ž3 are vertical
angles.
(Definition of
vertical angles)
13. mŽ3 5 908
(Definition of
vertical angles)
14. Ž3 is a right angle.
(Definition of right angle)
Geometry
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86
33. Given: h i k, j > h
34. Given: m > p, n > p
Prove: m i n
Prove: j > k
j
m
n
1
1
2
p
h
2
k
Statements (Reason)
1. m > p, n > p
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Statements (Reasons)
1. h i k, j > h
(Given)
2. Ž1 > Ž2
(Corresponding
Angles Postulate)
3. mŽ1 5 mŽ2
(Definition of
Congruent Angles)
2. Ž and Ž2 are right angles.
(Perpendicular lines intersect to
form four right angles.)
3. Ž1 > Ž2
(Right Angle
Congruence Theorem)
4. m i n (Corresponding Angles
Converse Postulate)
4. Ž1 is a right angle.
(Perpendicular lines intersect
35. no
to form four right angles.)
37. yes
5. mŽ1 5 908
(Definition of
right angle)
6. mŽ2 5 908
(Given)
36. no
38. yes
(Substitution)
7. Ž2 is a right angle. (Definition
of right angle)
8. j > k
(Definition of
perpendicular lines)
Geometry
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87
Mixed Review of
Problem Solving
2. 15;
1
5
1. a. y 5 35x, y 5 20x;
Cost (dollars)
y
150
skating
rink
100
50
0
bowling
alley
0 1 2 3 4 5 x
Number of hours
b. No; they don’t have the
same slope.
c. cost per hour of renting
d. y 5 20x 1 25;
y
150
Cost (dollars)
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
each facility
0
Sample answer: 2y 2 3x 5 22
4. 96.6. Sample answer: It’s the
perpendicular distance to the line.
5. a. 340 ft
17
b. }
25
100
50
3. Sample answer: 4x 1 6y 5 8;
c. Monongahela Incline.
bowling
alley
Sample answer: 0.7 . 0.68
0 1 2 3 4 5 x
Number of hours
Yes; y 5 20x and
y 5 20x 1 25 have the
same slope.
Geometry
Answer Transparencies for Checking Homework
88