Download M2 Geometry – Assignment sheet for Unit 2 Lines and Angles

M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
NAME
DATE
M2 Geometry – Assignment sheet for Unit 2 Lines and Angles, Packet 3
Unit 2 includes the following sections: 1-4, 1-5, 2-8, 1-6, 6-1, 3-1 to 3-6
Due
#
Assignment
Topics
2G
p. 176
# 13-18 all, 22-30 even
(In #13-18, you may assume that any
objects that look parallel actually are
parallel.)
2H
p. 7 in this packet
2I
p. 10 in this packet
3-1:
Vocabulary: parallel lines, skew lines, parallel
planes, transversal, consecutive interior angles,
alternate interior angles, alternate exterior
angles, corresponding angles
Identify angle pair relationships involving lines
and a transversal
3-2:
Investigate angle pair relationships involving
parallel lines and a transversal
Use algebra to solve problems involving parallel
lines and a transversal
3-5:
Use angle pair relationships to determine
whether lines are parallel
Test on 1-4, 1-5, 2-8, 1-6, 6-1, 3-1, 3-2, 3-5
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
Parallel Lines and Transversals
When two lines in the same plane do not intersect, they are parallel.
Lines that do not intersect and are not coplanar are skew.
In the figure above right,  is parallel to m , or   m . Arrows are drawn on the lines (not the
ones at the ends) to indicate that they are parallel. You can also write PQ  RS .
Similarly, if two planes do not intersect, they are parallel planes.
Use the figure at the right to identify each of the following.
1. a plane parallel to plane OPT
2. all segments parallel to NU
3. all segments that intersect MP
A transversal is a line that intersects two or more other lines at
two different points in a plane.
In the figure at right, line t is a transversal.
Two lines and a transversal form eight angles. Some pairs of
angles have special names. The next two pages show the
pairs of angles and their names.
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
4. Interior Angles are between the lines. Think of lines m and
n as the pieces of bread in a sandwich and the
transversal t as a toothpick holding the sandwich
together. Interior angles are the angles inside the
sandwich.
List the interior angles in this diagram:
∠ ____, ∠ ____, ∠ ____, and ∠ ____
5. Alternate interior angles are pairs of angles between two lines but on opposite sides of
the transversal. In other words, they are inside the sandwich but on opposite sides of the
toothpick.
There are two pairs of alternate interior angles in this diagram. Name them:
∠ ____ and ∠ ____ form a pair of alternate interior angles.
∠ ____ and ∠ ____ form a pair of alternate interior angles.
6. Consecutive interior angles are between two lines and on the same side of the
transversal. They are also called “same side interior”. In other words, they are inside the
sandwich and on the same side of the toothpick.
There are two pairs of consecutive interior angles in this diagram. Name them:
∠ ____ and ∠ ____ form a pair of consecutive interior angles.
∠ ____ and ∠ ____ form a pair of consecutive interior angles.
7. Alternate exterior Angles are outside the sandwich and on opposite sides of the
toothpick.
There are two pairs of alternate exterior angles in this diagram. Name them:
∠ ____ and ∠ ____ form a pair of alternate exterior angles.
∠ ____ and ∠ ____ form a pair of alternate exterior angles.
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
8. Corresponding angles are pairs of angles that are in the
same position in relation to one of the two lines and the
transversal.
For example, ∠1 and ∠5 form a pair of corresponding
angles because they are both above a piece of bread
and on the left side of the toothpick.
Name the other 3 pairs of corresponding angles:
∠ ____ and ∠ ____ form a pair of corresponding angles.
∠ ____ and ∠ ____ form a pair of corresponding angles.
∠ ____ and ∠ ____ form a pair of corresponding angles.
9. For each pair of angles: (i) Name the two lines (sandwich) and the transversal (toothpick)
that form the angles. (ii) Classify as alternate interior, consecutive interior, alternate
exterior, or corresponding.
a. ∠10 and ∠16
2 lines:
transversal:
Classification:
b. ∠4 and ∠12
2 lines:
transversal:
Classification:
c. ∠12 and ∠13
2 lines:
transversal:
Classification:
d. ∠3 and ∠9
2 lines:
transversal:
Classification:
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
ANGLES AND PARALLEL LINES
1. Open the applet at https://www.geogebra.org/m/P3nNeHjp. This link is available on the
class website.
2. The diagram in this applet shows two parallel lines cut by a transversal. Name these parts
below:
______  ______. The transversal is ______.
3. As usual, these 3 lines form 8 angles. How many different angle measures do you see?
4. Change the angle of the transversal by dragging point D or point E to the side. Make sure
that point F stays between A and B and that point G stays between C and H. How many
different angle measures do you see?
5. Suppose the angles on your screen are numbered
as shown at right.
a. Name the pairs of alternate interior angles in
the diagram at right. What do you notice
about the measures of the alternate interior angles
formed by the parallel lines in the applet?
b. Name the pairs of consecutive interior angles in the diagram above. What do you
notice about the measures of the consecutive interior angles formed by the parallel
lines in the applet?
c. Name the pairs of alternate exterior angles in the diagram above. What do you notice
about the measures of the alternate exterior angles formed by the parallel lines in the
applet?
d. Name the pairs of corresponding angles in the diagram above. What do you notice
about the measures of the corresponding angles formed by the parallel lines in the
applet?
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
e. Fill in the blanks:
When two parallel lines are cut by a transversal, they form…
…alternate interior angles that are _____________________,
…consecutive interior angles that are _____________________,
…alternate exterior angles that are ______________________, and
…corresponding angles that are _________________________.
 
6. In the figure at right, AH  EF . Fill in the blanks.
a. ∠CBH ≅ ∠FCB because they are ______________________________________ angles.
b. ∠DBH ≅ ∠BCG because they are ______________________________________ angles.
c. ∠FCE ≅ ∠BCG because they are ____________________________________ angles.
d. ∠DBA ≅ ∠GCE because they are ____________________________________ angles.
e. ∠HBC and ∠GCB are supplementary because they are _____________________ angles.
f.
∠ABC and ∠BCF are supplementary because they are _____________________ angles.
51° . Find the measure of each angle,
7. Suppose m∠ABC =
and explain why your answer is correct.
a. m∠BCG =
________ because
b. m∠FCE =
________ because
c. m∠BCF =
________ because
d. m∠GCE =
________ because
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
Assignment 2H:
1. In the figure, m∠9 = 80° and m∠5 = 68° . Find the measure of each angle.
a. ∠12
d. ∠3
b. ∠1
e. ∠7
c. ∠4
f. ∠16
2. Find the value of the variables in each figure. Show your work.
a.
b.
c.
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
PROVING LINES PARALLEL
If 2 lines are cut by a transversal and…
•
•
•
•
•
Then
corresponding ∠ s are ≅ ,
alternate exterior ∠ s are ≅ ,
consecutive interior ∠ s are supplementary,
alternate interior ∠ s are ≅ , or
⊥ to the transversal,
the lines are parallel.
Ex 1: If m∠1 = m∠2 , determine which lines, if any, are parallel. Explain your reasoning.
Solution: ∠1 and ∠2 are corresponding angles formed by
lines r and s with transversal m. Since ∠1 ≅ ∠2 , then r  s .
Ex 2: Find m∠ABC so that m  n . Explain your reasoning, and show your work.
Solution: m  n if alternate interior angles are congruent. Then
∠DAB ≅ ∠ABC
3 x + 10 = 6 x − 20
10
= 3 x − 20
30 = 3 x
x = 10
m∠ABC = 6 (10 ) − 20 = 40°
Find x so that   m . Explain your reasoning, and show your work.
1.
2.
3.
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
4. In the diagram at right, ∠1 ≅ ∠2 and ∠1 ≅ ∠3 . Explain why AB  DC .
5. In the diagram at right, ∠1 ≅ ∠5 and ∠15 ≅ ∠5 . Explain why   m and r  s .
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
Assignment 2I:
Determine which lines, if any, must be parallel if the given information is true. Explain your reasoning. Several
copies of the same diagram have been provided so you can draw on them.
1. ∠1 ≅ ∠2
2. ∠2 ≅ ∠9
3. ∠5 ≅ ∠7
4. m∠3 + m∠6 =180
5. ∠3 ≅ ∠7
6. ∠4 ≅ ∠5
7. If ∠1 ≅ ∠3 and AC  BD in the diagram at right, explain why AB  CD .
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
Practice Problems for Sections 3-1, 3-2, 3-5
1. Use the figure at the right to identify each of the following. You may assume that
segments appearing to be parallel actually are parallel in this diagram.
a. all planes that are parallel to plane DEH
b. all segments that are parallel to AB
c. all segments that are skew to CD
2. Identify the two lines and transversal forming each pair of angles.
Then classify the relationship between each pair of angles.
a. ∠ 4 and ∠ 10
d. ∠ 2 and ∠ 12
b. ∠ 7 and ∠ 3
e. ∠ 13 and ∠ 10
c. ∠ 8 and ∠ 14
f. ∠ 6 and ∠ 14
3. In the figure at right, m∠3 = 75° and m∠10 = 110° . Find the measure of each angle.
a. ∠ 2
d. ∠ 5
b. ∠ 7
e. ∠ 15
c. ∠ 14
f. ∠ 9
g. Is w  x ? Explain.
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
4. Find the value of the variable(s) in each figure. Explain your reasoning, and show your work.
a.
b.
5. Determine which lines, if any, must be parallel if the given information is
true. Explain your reasoning.
a. ∠3 ≅ ∠7
b. ∠10 ≅ ∠12
c. ∠2 ≅ ∠16
d. m∠5 + m∠12 =180
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
6. Find x so that   m . Explain your reasoning, and show your work.
7. If ∠1 and ∠2 are complementary and BC ⊥ CD , explain why BA  CD .
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
Review for 1-4, 1-5, 1-6, 2-8, 3-1, 3-2, 3-5
Use a protractor to find the measure of each angle to the nearest degree.
1.
2.
3.
4.
Figures on the rest of this review are not drawn to scale.
Use the diagram of ∠PQS at right to solve problems 5-6.
5. m∠PQS = 4 x, m∠SQR = 2 x, m∠RQP = 24° . Find m∠SQR and m∠PQS .
6. m∠SQR =
3 x − 2, m∠SQP =
5 x, m∠PQR =
34° . Find m∠SQR and m∠PQS .
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
7.

BD is the bisector of ∠ABC . m∠ABD=
m∠ABC =
____
( 2 y − 3) ° , and
m∠DBC =
____
m∠DBC =( y + 12 ) ° .
m∠ABD =
____

10 x . Find m∠4 .
8. PD is an angle bisector of ∠BPE . m∠4 = 4 x + 12 , and m∠BPE =
In problems 9-12, write an equation, and solve.
9. An angle is 40° more than its complement. What is the measure of that angle?
10. The measure of one angle is three times its complement. Find the measure of both angles.
11. The measure of an angle is 30 more than twice the measure of its supplement. Find the measure of both
angles.
12. An angle is 64 ° less than its supplement. What is the measure of this angle?
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
In problems 13-16, find the missing values of each letter.
13.
14.
15.
16.
In problems 17-20, find the values of x and y.
17.
18.
19.
20.
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
 
21. In the figure at right, TM ⊥ RS , and m∠QMS =°
58 . Find the measure of each
a. m∠TMQ =
____
b. m∠RMP =
____
c. m∠SMP =
____
d. m∠PMT =
____
T
M
R
P
3 x + 5 and m∠DEF =
2 x − 15 .
22. In the figure at right, m∠AEC =
Find each measure:
m∠DEF =
____
m∠DEB =
____
m∠CEB =
____
23. Complete the proof.
Given: ∠ABD ≅ ∠CBE
Prove: ∠ABE ≅ ∠CBD
Statements
Reasons
1.
1. Given
m∠CBE
2. m∠ABD =
2. ≅ ∠ s have = measures
3. m∠ABE + m∠EBD = m∠ABD
_______ + _______ =
m∠CBE
3.
4. m∠ABE + m∠EBD = m∠CBD + m∠EBD
4. Substitution
m∠EBD
5. m∠EBD =
5.
m∠CBD
6. m∠ABE =
7. ∠ABE ≅ ∠CBD
6.
7.
17
angle.
Q
S
M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
24. Complete the proof.
Given: ∠ABE ≅ ∠CBD
Prove: ∠ABD ≅ ∠CBE
Statements
Reasons
1. ∠ABE ≅ ∠CBD
1. Given
2.
2. ≅ ∠ s have = measures
3.
3. Reflexive
4. m∠ABE + m∠EBD = m∠CBD + m∠EBD
4. Addition
5.
5. Angle Addition
m∠CBE
6. m∠ABD =
6.
7.
7.
25. Complete the proof.
Given: ∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3
Statements
Reasons
1.
1. Given
2. m∠2 = m∠4
2. ≅ ∠ s have = measures
3. m∠1 + m∠2 =____
____ + ____ =
____
3.
4. m∠1 + m∠
=
2 ____ + ____
4. Transitive
5. m∠1 = m∠3
5.
6.
6. ≅ ∠ s have = measures
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
26. Complete the proof.
Given: AB ⊥ DC , ∠1 ≅ ∠4
Prove: ∠3 ≅ ∠2
Statements
Reasons
1. AB ⊥ DC , ∠1 ≅ ∠4
1. Given
2. m∠1 = m∠4
2. ≅ ∠ s have = measures
3. m∠ABC = 90°, m∠ABD = 90°
3.
m∠ABD
4. m∠ABC =
4.
5. ______ + ______ = ______
______ + ______ = ______
5. Angle Addition
6. m∠1 + m∠2 = m∠3 + m∠4
6.
7. m∠2 = m∠3
7.
8. ∠2 ≅ ∠3
8.
9. ∠3 ≅ ∠2
9.
27.Sketch a concave hexagon.
28. Sketch a convex pentagon.
29. Sketch a convex regular heptagon, and mark the sides and/or angles with tic marks and/or arcs as
appropriate.
30. Sketch a concave equilateral nonagon, and mark the sides and/or angles with tic marks and/or arcs as
appropriate.
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
31. Find the sum of the measures of the interior angles of a convex polygon with 13 sides.
32. Find the sum of the measures of the interior angles of a convex polygon with 15 sides.
33. The sum of the measures of the interior angles of a convex polygon is 7020° . Find the number of sides of
the polygon.
34. The sum of the measures of the interior angles of a convex polygon is 1980° . Find the number of sides of
the polygon.
35. Find the measure of each interior angle of a regular heptagon.
36. Find the measure of each interior angles of a regular nonagon.
37. Find the measure of each exterior angle of a regular decagon.
38. Find the measure of each exterior angle of a regular 18-gon.
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
In #39-42, find the value of x.
39.
40.
41.
42.
43. How many sides does a regular polygon have if each interior angle measures 168.75° ?
44. How many sides does a regular polygon have if each interior angle measures 150° ?
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
In #45-48, refer to the figure at the right to identify each of the following.
45. all planes that intersect plane STX
����
46. all segments that intersect 𝑄𝑄𝑄𝑄
����
47. all segments that are parallel to 𝑋𝑋𝑋𝑋
�����
48. all segments that are skew to 𝑉𝑉𝑉𝑉
In #49-54, classify the relationship between each pair of angles as alternate interior, alternate exterior,
corresponding, or consecutive interior angles.
49. ∠ 2 and ∠ 10
50. ∠ 7 and ∠ 13
51. ∠ 9 and ∠ 13
52. ∠ 6 and ∠ 16
53. ∠ 3 and ∠ 10
54. ∠ 8 and ∠ 14
In #55-58, name the transversal that forms each pair of angles.
Then identify the special name for the angle pair.
55. ∠ 2 and ∠ 12
56. ∠ 6 and ∠ 18
57. ∠ 13 and ∠ 19
58. ∠ 11 and ∠ 7
For #59-64, m ∠ 2 = 92 ° and m ∠ 12 = 74 ° . Find the measure of each angle.
59. ∠ 10
60. ∠ 8
61. ∠ 9
62. ∠ 5
63. ∠ 11
64. ∠ 13
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M2 GEOMETRY PACKET 3 FOR UNIT 2 – SECTIONS 3-1, 3-2, 3-5
In #65-66, find the value of the variable(s) in each figure.
65.
66.
In #67-70, given the following information, determine which lines, if any, must be parallel. State the
reason that justifies your answer.
67. m ∠ BCG + m ∠ FGC = 180
68. ∠ CBF ≅ ∠ GFH
69. ∠ EFB ≅ ∠ FBC
70. ∠ ACD ≅ ∠ KBF
In #71-72, find x so that   m .
71.
72.
73. If ∠2 and ∠3 are supplementary, explain why AB  CD .
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