Download Investigating Geometry Activity: The Transitive Property of Parallel

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3.3
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Investigating Geometry Activity:
The Transitive Property of Parallel Lines
For use before the lesson “Prove Lines are Parallel”
Materials: Question
explore
straightedge and tracing paper
What is the transitive property of parallel lines?
Investigating Parallel Lines
STEP 1 Draw Parallel Lines
Place your straightedge on your paper and trace
both sides to draw parallel lines. Label the first line
as “line 1” and the second line as “line 2.”
STEP 2 Draw Third Line
Place your straightedge on your paper so that
the top edge is lined up with line 2. Now trace the
other side of the straightedge to draw a line that is
parallel to line 2. Label this line as “line 3.”
Line 1
Line 2
1
2
3
4
5
6
Line 3
STEP 4 Label Angles
Label the angles formed by the transversal and
line 1 as shown. Now, label the angles formed
by the transversal and line 3 as shown.
STEP 5 Compare Angles
Place your tracing paper on top of angles a, b, c,
and d. Trace these angles. Slide the tracing paper
down so that the traced angles are on top of
angles e, f, g, and h. Compare the angles.
draw
conclusions
a b
c d
Line 1
Line 2
e f
g h
Line 3
Use your observations to complete the following.
1. How do the corresponding angles b and f compare?
Lesson 3.3
2. How do the alternate interior angles c and f compare?
3-32
3. How do the alternate exterior angles a and h compare?
4. You drew lines 1, 2, and 3 so that line 1 is parallel to line 2, and line 2 is parallel
to line 3. However, is line 1 parallel to line 3? Explain your answer.
5. Complete the statement: If two lines are parallel to the same line, then they
are    ?    to each other.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
STEP 3 Draw Transversal
Use your straightedge to draw a nonperpendicular transversal to all three lines.
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C3L03IG.indd 32
4/27/11 2:45:35 AM
Lesson 3.2 Use Parallel Lines
and Transversals, continued
B
Study Guide
1. Using the Vertical Angles Congruence
Theorem, m∠ 8 5 658. By the Corresponding
Angles Postulate, m∠ 4 5 658. Because ∠ 8
and ∠ 6 are corresponding angles, by the
Corresponding Angles Postulate, you know that
m∠ 6 5 658. 2. Using the Vertical Angles
Congruence Theorem, m∠ 3 5 1158. By the
Corresponding Angles Postulate, m∠ 7 5 1158.
Because ∠ 3 and ∠ 1 are corresponding angles,
by the Corresponding Angles Postulate, you know
that m∠ 1 5 1158. 3. 68 4. 25 5. 12 6. 10
7. 10 8. 5 9. 12 10. 16
Problem Solving Workshop:
Using Alternative Methods
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
1. 1158; by the Alternate Exterior Angles
Theorem 2. 308; by the Consecutive Interior
Angles Theorem
Challenge Practice
1. m∠ 1 5 428, m∠ 2 5 1388, m∠ 3 5 1388,
m∠ 4 5 428, m∠ 5 5 1328, m∠ 6 5 488,
m∠ 7 5 488, m∠ 8 5 1328, m∠ 9 5 908,
m∠ 10 5 908, m∠ 11 5 908, m∠ 12 5 908,
m∠ 13 5 1328, m∠ 14 5 488, m∠ 15 5 488,
m∠ 16 5 1328, m∠ 17 5 428, m∠ 18 5 1388,
m∠ 19 5 1388, m∠ 20 5 428
2. m∠ 1 5 358, m∠ 2 5 1458, m∠ 3 5 1118,
m∠ 4 5 698, m∠ 5 5 1118, m∠ 6 5 698,
m∠ 7 5 1458, m∠ 8 5 358, m∠ 9 5 698,
m∠ 10 5 1118, m∠ 11 5 698, m∠ 12 5 1118,
m∠ 13 5 768, m∠ 14 5 1048, m∠ 15 5 768,
m∠ 16 5 1048, m∠ 17 5 1048, m∠ 18 5 768,
m∠ 19 5 1048, m∠ 20 5 768
3. m∠ 1 5 1008, m∠ 2 5 808, m∠ 3 5 808,
m∠ 4 5 1008, m∠ 5 5 1008, m∠ 6 5 568,
m∠ 7 5 248, m∠ 8 5 248, m∠ 9 5 568,
m∠ 10 5 1008, m∠ 11 5 1568, m∠ 12 5 248,
m∠ 13 5 248, m∠ 14 5 1568, m∠ 15 5 1248,
m∠ 16 5 568, m∠ 17 5 1248, m∠ 18 5 568,
m∠ 19 5 1008, m∠ 20 5 808, m∠ 21 5 1008,
C
answers
Theorem; Corresponding Angles Postulate;
­Transitive Property of Congruence
20. Given; Alternate Exterior Angles Theorem;
Given; Corresponding Angles Postulate; Transitive
Property of Congruence
m∠ 22 5 808, m∠ 23 5 1568, m∠ 24 5 248,
m∠ 25 5 248, m∠ 26 5 1568, m∠ 27 5 1008,
m∠ 28 5 568, m∠ 29 5 248, m∠ 30 5 248,
m∠ 31 5 568, m∠ 32 5 1008
4. Sample answer:
A
D
Statements
Reasons
} } } }
​  i DC​
​  , AD​
​  i BC​
​   1. Given
1. AB​
2. ∠ A and ∠ B and 2. Consecutive
∠ B and ∠ C Interior
are supplementary. Angles Theorem
3. m∠ A 1 m∠ B 5 1808, 3. Definition of
m∠ B 1 m∠ C 5 1808 supplementary
angles
4. m∠ A 1 m∠ B 5 4. Substitution
m∠ B 1 m∠ C Property of Equality
5. m∠ A 5 m∠ C 5. Subtraction
Property of Equality
6. ∠ A > ∠ C 6. Definition of
congruent angles
5. x 5 67; Draw a line through the angle x8 that
is parallel to both m and n. Then using the
Alternate Interior Angles Theorem and the
definition of supplementary angles, you can
determine that x 5 35 1 32 5 67.
Lesson 3.3 Prove Lines are
Parallel
Teaching Guide
Diagram 2; the lines appear to never intersect.
1. ∠12 and ∠14; ∠11 and ∠13; the angles in
each pair have the same measure. 2. ∠4 and ∠6;
∠3 and ∠5; the angles in each pair have different
measures. 3. If the measures of the alternate
interior angles are equal, the lines are parallel.
4. ∠9 and ∠16; ∠10 and ∠15; the angles in
each pair have the same measure. 5. ∠1 and ∠7;
∠2 and ∠8; the angles in each pair have different
measures. 6. If the measures of the alternate
exterior angles are equal, the lines are parallel.
Investigating Geometry Activity
1. They are congruent. 2. They are congruent.
3. They are congruent. 4. Yes, line 1 is parallel
to line 3, because corresponding angles are
Geometry
Chapter Resource Book
A35
c­ ongruent, alternate interior angles are congruent,
and alternate exterior angles are congruent.
5. If two lines are parallel to the same line, then
they are parallel to each other.
Practice Level A
1. yes; Corresponding Angles Converse
13. p i q 14. neither 15. Given; Alternate
Exterior Angles Theorem; Definition of Congruent
Angles; Given; Definition of Congruent Angles;
Alternate Interior Angles Converse 16. Given;
Alternate Interior Angles Theorem; Definition of
Congruent Angles; Given; Substitution; Definition
of Supplementary Angles; Consecutive Interior
Angles Converse
2. yes; Alternate Interior Angles Converse
Study Guide
3. yes; Alternate Exterior Angles Converse
1. 18 2. Yes; the angle that corresponds with
708 has a measure of 708 because it is a linear pair
with the angle that measures 1108.
3. 11 4. 8 5. 12
4. yes; Corresponding Angles Converse
5. no 6. yes; Alternate Interior Angles Converse
7. 40 8. 30 9. 30 10. 14 11. 32 12. 95
13. C 14. m i n 15. p i q 16. p i q
17. neither 18. Given; Corresponding Angles
Postulate; Given; Transitive Property of
Congruence; Corresponding Angles Converse
19. Each lane is parallel to the one next to it,
so l1 i l2, l2 i l3, and l3 i l4. Then l1 i l3 by
the Transitive Property of Parallel Lines. By
continuing this reasoning, l1 i l4. So, the first
lane is parallel to the last lane.
Practice Level B
1. yes; Consecutive Interior Angles Converse
2. yes; Alternate Interior Angles Converse
3. no 4. 40 5. 109 6.115 7. 22 8. 5 9. 80
10. congruent 11. supplementary 12. congruent
13. Each row is parallel to the one next to it,
so r1 i r2, r2 i r3, and so on. Then r1 i r3 by
the Transitive Property of Parallel Lines.
By continuing this reasoning, r1 i r5. So, the first
row is parallel to the last row. 14. Given
15. Corresponding Angles Postulate 16. Given
17. Transitive Property of Congruence 18. Alternate Exterior Angles Converse 19. Given
20. Alternate Interior Angles Theorem
21. Given 22. Transitive Property of Congruence
23. Alternate Interior Angles Converse
24. Corresponding Angles Converse
Practice Level C
1. no 2. no 3. yes; Alternate Exterior Angles
Converse 4. 16 5. 15 6. 24 7. 45 8. 23
A36
9. 42 10. m i n 11. neither 12. neither
Geometry
Chapter Resource Book
Problem Solving Workshop:
Mixed Problem Solving
@##$ , @​BF ​
##$ 
1. a. Sample answer: ​CG​
@##$ c. Plane DCG
b. Sample answer: ​CG​
2. Yes. Since line m intersects line j, the angle
formed that is 498 and the one below it are
supplementary. So, the angle below it is 1318. Since
line n intersects line j, the angle formed that is 1318
and the one above it are supplementary. So, the
angle above it is 498. Since a pair of corresponding
angles are congruent, the lines m and n are parallel.
3. 1108 4. Sketches will vary. There are 6 sides
that could be a transversal for a pair of opposite
sides. 5. a. no b. yes c. Line l on any step is
always parallel to line l on any other step because
the line is always going across the width of the
escalator. Plane A on any step is not always parallel
to plane A on any other step. Plane A on the step
that is rising up the escalator is not parallel to
plane A on the step coming down underneath it.
d. 2; When each step is going from facing upward
to facing downward and when each step is going
from facing downward to facing upward
Challenge Practice
1. a. ∠ VXW 5 (180 2 x)8
b. ∠ WXZ 5 (180 2 y)8
c. ∠ VXZ 5 (360 2 x 2 y)8
2. p i q by the Corresponding Angles Converse;
q i r by the Consecutive Interior Angles Converse;
p i r by the Transitive Property of Parallel Lines;
s i t by the Alternate Exterior Angles Converse
3. x 5 6, y 5 9
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
answers
Lesson 3.3 Prove Lines are
Parallel, continued