Download Investigating Parallel Lines and Angle Pairs Key

Geometry
HS Mathematics
Unit: 03 Lesson: 01
Investigating Parallel Lines and Angle Pairs KEY
Lines m and n are parallel. Eight angles are formed when transversal t intersects lines m and n. The
following postulate and theorems represent the relationships between the angles formed when
parallel lines are cut by a transversal.
t
m
n
1
2
3
4
5
6
7
8
Corresponding Angle Postulate: If two parallel lines are cut by a transversal, then the
corresponding angles are congruent.
∠1 ≅ ∠5 ∠2 ≅ ∠6 ∠3 ≅ ∠7 ∠4 ≅ ∠8
Alternate Interior Angle Theorem: If two parallel lines are cut by a transversal, then the alternate
interior angles are congruent.
∠3 ≅ ∠6
∠4 ≅ ∠5
Same Side Interior Angle Theorem: If two parallel lines are cut by a transversal, then consecutive
(same side) interior angles are supplementary.
∠3 is supplementary to ∠5
∠ 4 is supplementary to ∠6
Same Side Exterior Angle Theorem: If two parallel lines are cut by a transversal, then same side
exterior angles are supplementary.
©2012, TESCCC
07/19/12
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∠1 is supplementary to ∠7
∠2 is supplementary to ∠8
Geometry
HS Mathematics
Unit: 03 Lesson: 01
Alternate Exterior Angle Theorem: If two parallel lines are cut by a transversal, then alternate
exterior angles are congruent.
∠1 ≅ ∠8
∠2 ≅ ∠7
Perpendicular Transversal Theorem: If a line is perpendicular to one of two parallel lines, then it is
perpendicular to the second line.
©2012, TESCCC
07/19/12
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Geometry
HS Mathematics
Unit: 03 Lesson: 01
Investigating Parallel Lines and Angle Pairs KEY
Guided Practice
p
r
s
q
1
3
4
5
6
8
7
2
16
15
14
13
12
11
10
9
62o
62o
62o
62o
118o
118o
118o
118o
90o
90o
90o
90o
90o
90o
90o
90o
Given r  q.
©2012, TESCCC
07/19/12
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Geometry
HS Mathematics
Unit: 03 Lesson: 01
1. Name four pair of corresponding angles.
Answers will vary. Sample: 1 & 8, 2 & 5, 3 & 6, 12 & 16
2. Name four pair of alternate interior angles.
Answers will vary. Sample: 3 & 8, 4 & 5, 11 & 14, 12 & 13
3. Name four pair of same side interior angles.
Answers will vary. Sample: 3 & 5, 4 & 8, 11 & 13, 12 & 14
4. Name four pair of alternate exterior angles.
Answers will vary. Sample: 2 & 7, 1 & 6, 9 & 16, 10 & 15
∠1 = 62o
5. Given
, find all the measure of all the other angles.
See diagram above.
6. What can be said about line p and line q. Justify this with a theorem.
Line p is perpendicular to line q.
If a line is perpendicular to one of two parallel lines, then it is perpendicular to the second line.
©2012, TESCCC
07/19/12
page 4 of 7
Geometry
HS Mathematics
Unit: 03 Lesson: 01
Investigating Parallel Lines and Angle Pairs KEY
Practice Problems
1. State the postulate or theorem that allows the conclusion that if a  b, then
∠1 ≅ ∠2
.
1
2
a
b
1
2
a
b
1
2
a
b
a.
b.
a. Alt. ext. angle thm
2. In the figure below m  n,
the other angles.
c.
b. Corr. Angle post.
AC

BD
c. Alt. int. angle thm.
, and the measure of
∠1 = 148o
. Find the measure of all
A
B
C
D
1
7
6
5
4
3
2
8
m
n
148o
©2012, TESCCC
07/19/12
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Geometry
HS Mathematics
Unit: 03 Lesson: 01
Obj110
148
148o
148o
32o
32o
32o
32o
148o
3. In the figure k  l and s  t. Find the values of x, y, and z and justify by theorem.
k
l
t
s
98o
14zo
(2x+5)o
(3y+8)o
x = 38.5, y = 30, z = 7
x – Same-side int. angle thm.
y – Alt. int. angle thm.
z – Alt. int. angle thm.
©2012, TESCCC
07/19/12
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Geometry
HS Mathematics
Unit: 03 Lesson: 01
Investigating Parallel Lines and Angle Pairs KEY
suur
AB
suur
AB
suur
TP
suur
AB
4. Draw
and Point T not on
. Construct
parallel to
through Point T. Justify that
suur
suur
AB
TP
and
are parallel using the relationships between angle pairs.
See the construction.
The corresponding angles are constructed to be congruent, so the lines are parallel.
5. Explain how to create two parallel lines with a perpendicular transversal. Construct the
situation. Compare and contrast the relationships between the angle pairs created.
Answers may vary.
Sample: Draw a line with a given point on the line. Construct a perpendicular line through that
point by marking equidistant arcs on the line on either side of the point. Using the intersection
points of the line and two arcs, mark equidistant, intersecting arcs above the line. Draw a line
through the given point on the line and the intersection of the arcs above the line. This should
be a line perpendicular to the original line. Place a point on this perpendicular line. Construct
congruent angles using this point and the original given point to draw the parallel line to the
original line.
See the construction.
All angles are equal to 90o, therefore angle pairs are either congruent and supplementary.
©2012, TESCCC
07/19/12
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