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Crisscross
Applesauce
Angle Relationships Formed
by Two Lines Intersected
by a Transversal
Learning Goals
Key Terms
In this lesson, you will:
 transversal
 alternate interior angles
 Explore the angles determined by two lines that are intersected by a transversal.
 Explore the measures of angles determined by two parallel lines that are intersected by a transversal.
 alternate exterior angles
 same-side interior angles
 same-side exterior angles
 Identify alternate interior angles.
 Identify alternate exterior angles.
 Identify same-side interior angles.
 Identify same-side exterior angles.
 Identify corresponding angles.
 Determine the measure of alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles, and corresponding angles.
T
ake two straws and lay them on your desk. Make them as close to parallel as © 2011 Carnegie Learning
© 2011 Carnegie Learning
you can. Then lay a third straw on top of the other two at any angle you like. Tape your entire construction together.
Use your protractor to measure the angles you see. Notice anything interesting? Compare your constructions with your classmates’ constructions. What do you notice?
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 545
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 545
Problem 1
Transversal, alternate interior
angles, alternate exterior angles,
same-side interior angles,
same-side exterior angles,
and corresponding angles
are defined. Students sketch
examples of these special pairs
of angles and answer questions
related to each pair of angles.
Materials
Protractor
Straightedge
Problem 1
Naming All the Angles
In this lesson, you will explore all the angles that can be formed by transversals.
A transversal is a line that intersects two or more lines.
1. Sketch an example of a transversal.
2. Compare your sketch with your classmates’ sketches. Did everyone sketch the same
figure? Explain how the sketches are the same or different.
No. Everyone did not sketch the same figure. Some students sketched more than
three lines and some sketches did not contain any parallel lines. Each of the sketches
shows at least three lines with at least one of the lines intersecting two or more lines.
Grouping
Have students complete
Questions 1 through 8 with
a partner. Then share the
responses as a class.
Alternate interior angles are angles formed when a line (transversal) intersects two
other lines. These angles are on opposite sides of the transversal and are between the
other two lines.
3. Sketch an example of alternate interior angles.
1
Share Phase,
Questions 1 through 5
4. How many pairs of alternate interior angles are formed by two lines that are
intersected by a transversal?
Two pairs of alternate interior angles are formed by two lines that are intersected
by a transversal.
5. Compare your sketch with your classmates’ sketches. Did everyone draw the same
alternate interior angles? Explain how the sketches are the same or different.
© 2011 Carnegie Learning
Does a transversal always
intersect two parallel lines?
Explain.
2
angles formed have different measures, and some sketches do not contain parallel
lines. Each of the sketches shows angles on opposite sides of the transversal and
between the two other lines.
546 • Chapter 10 Line and Angle Relationships
546 • Chapter 10 Line and Angle Relationships
© 2011 Carnegie Learning
No. Everyone did not draw the same alternate interior angles. The alternate interior
Share Phase,
Questions 6 through 8
• What is the difference
between alternate interior
and alternate exterior
angles?
Alternate exterior angles are angles formed when a line (transversal) intersects two
other lines. These angles are on opposite sides of the transversal and are outside the
other two lines.
6. Sketch an example of alternate exterior angles.
• What do alternate interior
1
and alternate exterior angles
have in common?
2
7. How many pairs of alternate exterior angles are formed by two lines that are
intersected by a transversal?
Two pairs of alternate exterior angles are formed by two lines that are intersected
by a transversal.
8. Compare your sketch with your classmates’ sketches. Did everyone draw the same
alternate exterior angles? Explain how the sketches are the same or different.
No. Everyone did not draw the same alternate exterior angles. The alternate
exterior angles formed have different measures, and some sketches do not contain
parallel lines. Each of the sketches shows angles on opposite sides of the
transversal and outside the two other lines.
Grouping
other lines. These angles are on the same side of the transversal and are between the
other two lines.
9. Sketch an example of same-side interior angles.
© 2011 Carnegie Learning
Have students complete
Questions 9 through 17 with
a partner. Then share the
responses as a class.
Same-side interior angles are angles formed when a line (transversal) intersects two
1
2
10. How many pairs of same-side interior angles are formed by two lines that are
intersected by a transversal?
© 2011 Carnegie Learning
Two pairs of same-side interior angles are formed by two lines that are intersected
by a transversal.
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 547
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 547
Share Phase,
Questions 11 through 14
• What is the difference
between same-side interior
and same-side exterior
angles?
• What do same-side interior
and same-side exterior
angles have in common?
11. Compare your sketch with your classmates’ sketches. Did everyone draw the same
angles? Explain how the sketches are the same or different.
No. Everyone did not draw the same angles. The same-side interior angles formed
have different measures, and some sketches do not contain parallel lines. Each of
the sketches shows angles on the same side of the transversal and between the
two other lines.
Same-side exterior angles are angles formed when a line (transversal) intersects two
other lines. These angles are on the same side of the transversal and are outside the other
two lines.
12. Sketch an example of same-side exterior angles.
1
2
13. How many pairs of same-side exterior angles are formed by two lines that are
intersected by a transversal?
Two pairs of same-side exterior angles are formed by two lines that are
intersected by a transversal.
14. Compare your sketch with your classmates’ sketches. Did everyone draw the same
angles? Explain how the sketches are the same or different.
No. Everyone did not draw the same angles. The same-side exterior angles formed
have different measures, and some sketches do not contain parallel lines. Each of
two other lines.
Recall that corresponding angles are angles that have the same relative positions in
geometric figures.
15. Sketch an example of corresponding angles. Include two lines intersected by a
© 2011 Carnegie Learning
the sketches shows angles on the same side of the transversal and outside the
transversal in the sketch.
2
548 • Chapter 10 Line and Angle Relationships
548 • Chapter 10 Line and Angle Relationships
© 2011 Carnegie Learning
1
Share Phase,
Questions 15 through 17
16. How many pairs of corresponding angles are formed by two lines that are intersected
When a transversal intersects
two lines, how many pairs
of corresponding angles are
formed?
by a transversal?
Four pairs of corresponding angles are formed by two lines that are intersected by
a transversal.
17. Compare your sketch with your classmates’ sketches. Did everyone draw the same
corresponding angles? Explain how the sketches are the same or different.
No. Everyone did not draw the same corresponding angles. Some students drew
corresponding angles with different measures, and some sketches did not contain
parallel lines. Each of the sketches shows angles on the same side of the
transversal, and one angle is located between the two lines and one angle is
outside the two lines.
Problem 2
Problem 2
Students are given three
diagrams and will identify the
transversals.
Where Are the Transversals?
1. Suppose that <1 i <2, and both lines intersect <3. Identify the transversal(s).
Grouping
© 2011 Carnegie Learning
Share Phase,
Questions 1 through 3
• For a line to be considered
3
1
2
<3 is the transversal.
© 2011 Carnegie Learning
Have students complete
Questions 1 through 3 with
a partner. Then share the
responses as a class.
a transversal, the line has to
intersect how many other
lines?
2. Suppose that ,1 i ,2, and both lines intersect ,3. Identify the transversal(s).
3
1
2
<1, <2, and <3 are all transversals.
• If two coplanar non-parallel
lines are not fully extended
and you cannot see the point
of intersection, do they still
intersect?
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 549
• How many lines does ℓ2 intersect?
• How many lines does ℓ3 intersect?
• How many lines does ℓ1
intersect?
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 549
3. The arrowheads on these line segments indicate parallel relationships between
opposite sides of the geometric figure. Transversals can be lines or line segments.
Does this figure contain transversals? Explain your reasoning.
Thisgeometricfigurecontainsfourtransversals.Eachsideofthefigure
intersectstwootherlinesegments,soeachlinesegmentisatransversal.
Street Map of Atlantic City, New Jersey
Refer to the map of part of Atlantic City, New Jersey, to answer each question. Assume all
line segments that appear to be perpendicular are perpendicular. Assume all line
segments that appear to be parallel are parallel.
MA
NEW
.
AVE
.
AVE
DEL
RA
MO
MAR
INE
IRE
MA
PSH
HAM
NEW
NT
MO
VER
AND
ISL
DE
RHO
UT
CTIC
SEY
NNE
JER
CO
ARE
AW
AND
RYL
To New York
& Philadelphia
.
AVE
.
AVE
N
R
ITE
P.O.
.
AVE
.
AVE
P.O.
IC
IF
PAC
.
AVE
.
AVE
.
AVE
NTIC
A
ATL
.
AVE
.
AVE
.
AVE
TIC
ARC
Light House
.
AVE
.
UNT
MO
FAIR
.
AVE
12
34
.
AVE
AVE
.
AVE
.
AVE
.
AVE
.
AVE
.
AVE
Have students complete
Questions 1 through 5 with
a partner. Then share the
responses as a class.
56
78
.
AVE
MED
.
AVE
.
AVE
Grouping
EA
RAN
.
AVE
IA
GIN
ND
VIR
A
STR
A
THE
LIN
ARO
A
N. C
LIN
ARO
S. C
SEE
NES
TEN
K
YOR
NEW CKY
TU
KEN
OIS
ILLIN
A
IAN
IND
IO
OH
AN
HIG
MIC
K
AL
DW
AR
BO
N
W
E
S
Share Phase,
Questions 1 and 2
• Does Atlantic Avenue
intersect more than two other
avenues?
• How many distinct angles
are formed when two lines
intersect?
• Is there more than one
avenue you could have
chosen? Explain.
Atlantic City
N.J.
1. Is Atlantic Ave. a transversal? Explain your reasoning.
AtlanticAve.isatransversalbecauseitintersectsmorethantwootheravenues.
2. Locate the circle drawn on Atlantic Ave. This circle is drawn at the intersection of
Atlantic Ave. and what other avenue?
ThecircleisdrawnattheintersectionofAtlanticAve.andN.CarolinaAve.
550 • Chapter 10 Line and Angle Relationships
550 • Chapter 10 Line and Angle Relationships
© 2011 Carnegie Learning
Students are given a street map
of Atlantic City, N.J., and use
the map to identify examples of
transversals, alternate interior
angles, alternate exterior angles,
same-side interior angles,
same-side exterior angles, and
corresponding angles.
Problem 3
© 2011 Carnegie Learning
Problem 3
Share Phase,
Questions 3 through 5
• How many pairs of alternate
3. How many angles are formed at this intersection?
Four angles are formed at this intersection.
interior angles are formed?
• How many pairs of alternate
4. Label each angle.
exterior angles are formed?
a. Place a 1 on the angle that would be considered the northwest angle.
• How many pairs of same-
b. Place a 2 on the angle that would be considered the northeast angle.
c. Place a 3 on the angle that would be considered the southwest angle.
side interior angles are
formed?
d. Place a 4 on the angle that would be considered the southeast angle.
• How many pairs of same-
5. Using Atlantic Ave. and N. Carolina Ave., choose a third avenue such that Atlantic
Ave. is a transversal.
side exterior angles are
formed?
a. Label the four angles at this intersection /5, /6, /7, and /8 and describe the
location of each angle (northeast, northwest, southeast, or southwest).
• How many pairs of
corresponding angles are
formed?
Answers will vary.
Atlantic Ave. is a transversal between N. Carolina Ave. and New Jersey Ave.
/5 is northwest, /6 is northeast, /7 is southwest, and /8 is southeast.
b. List all pairs of alternate interior angles.
/5, /4 and /2, /7 are pairs of alternate interior angles.
c. List all pairs of alternate exterior angles.
/1, /8 and /3, /6 are pairs of alternate exterior angles.
d. List all pairs of same-side interior angles.
© 2011 Carnegie Learning
/2, /5 and /4, /7 are pairs of same-side interior angles.
e. List all pairs of same-side exterior angles.
/1, /6 and /3, /8 are pairs of same-side exterior angles.
© 2011 Carnegie Learning
f. List all pairs of corresponding angles.
/1, /5; /2, /6; /3, /7; and /4, /8 are pairs of corresponding angles.
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 551
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 551
Problem 4
Students are given a street map
of Washington, D.C., and use
the map to identify examples
of transversals, alternate interior
angles, alternate exterior angles,
same-side interior angles,
same-side exterior angles, and
corresponding angles.
Problem 4
Washington, D.C., Map
Use the map of Washington, D.C., to answer each question. Assume all line segments that
appear to be parallel are parallel.
Rhode Island Ave.
New York Ave.
Q St.
1 2
3 4
Grouping
could be labeled ∠1? ∠2?
∠3? ∠4?
• Did everyone place the label
6th St.
7th St.
9th St.
New Jersey Ave.
Massachusetts Ave.
1. Label /1, /2, /3, and /4 at the intersection of 7th St. and P St.
2. Label /5, /6, /7, and /8 at the intersection of 6th St. and P St.
3. Label /9, /10, /11, and /12 at the intersection of Massachusetts Ave. and P St.
4. Use a protractor to measure all 12 angles.
5. Consider only 6th St., 7th St., and P St.
a. Which of these streets, if any, are transversals?
P St. is a transversal.
for ∠5 in the same position?
• How many different positions
could be labeled ∠5? ∠6?
∠7? ∠8?
• Did everyone place the label
b. Name the pairs of alternate interior angles. What do you notice about their
angle measures?
for ∠9 in the same position?
/2 and /7
• How many different positions
/4 and /5
could be labeled ∠9? ∠10?
∠11? ∠12?
The alternate interior angles in each pair have equal measures.
• How did you determine
which street was a
transversal?
552 • Chapter 10 Line and Angle Relationships
• How did you locate the pairs
of alternate interior angles?
552 • Chapter 10 Line and Angle Relationships
© 2011 Carnegie Learning
• How many different positions
P St.
© 2011 Carnegie Learning
for ∠1 in the same position?
9 10
11 12
N St.
Have students complete
Questions 1 through 6 with
a partner. Then share the
responses as a class.
Share Phase,
Questions 1 through 5,
parts (a) and (b)
• Did everyone place the label
5 6
7 8
Share Phase,
Question 5, parts (c)
through (g)
• How did you locate the pairs
of alternate exterior angles?
c. Name the pairs of alternate exterior angles. What do you notice about their
angle measures?
/3 and /6
/1 and /8
• How did you locate the
The alternate exterior angles in each pair have equal measures.
corresponding angles?
• How did you locate the
same-side interior angles?
d. Name the pairs of corresponding angles. What do you notice about their
• How did you locate the
angle measures?
same-side exterior angles?
• Does 6th Street intersect 7th
/1 and /5
/2 and /6
Street?
/3 and /7
/4 and /8
The corresponding angles in each pair have equal measures.
e. Name the pairs of same-side interior angles. What do you notice about their
angle measures?
/2 and /5
/4 and /7
The same-side interior angles in each pair are supplementary.
f. Name the pairs of same-side exterior angles. What do you notice about their
angle measures?
© 2011 Carnegie Learning
/1 and /6
/3 and /8
The same-side exterior angles in each pair are supplementary.
g. What is the relationship between 6th St. and 7th St.?
© 2011 Carnegie Learning
6th St. is parallel to 7th St.
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 553
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 553
Share Phase,
Question 6
How did you determine which
of the three streets were
transversals?
6. Consider only 6th Street, Massachusetts Avenue, and P Street.
a. Which of these streets, if any, are transversals?
6th St., Massachusetts Ave., and P St. are all transversals.
b. Name the pairs of alternate interior angles. What do you notice about their
angle measures?
/6 and /11
/8 and /9
The alternate interior angles in each pair do not have equal measures.
c. Name the pairs of alternate exterior angles. What do you notice about their
angle measures?
/5 and /12
/7 and /10
The alternate exterior angles in each pair do not have equal measures.
d. Name the pairs of corresponding angles. What do you notice about their
angle measures?
/5 and /9
/6 and /10
/7 and /11
/8 and /12
The corresponding angles in each pair do not have equal measures.
e. Name the pairs of same-side interior angles. What do you notice about their
angle measures?
/6 and /9
/8 and /11
f. Name the pairs of same-side exterior angles. What do you notice about their
angle measures?
/5 and /10
/7 and /12
There is no special relationship.
© 2011 Carnegie Learning
There is no special relationship.
g. What is the relationship between 6th St. and Massachusetts Ave.?
554 • Chapter 10 Line and Angle Relationships
554 • Chapter 10 Line and Angle Relationships
© 2011 Carnegie Learning
6th St. and Massachusetts Ave. are intersecting lines that are not parallel.
Problem 5
Students use a protractor
to measure special pairs of
angles formed by a transversal
intersecting two non-parallel
lines and a transversal
intersecting two parallel lines.
They conclude that when two
parallel lines are intersected
by a transversal the alternate
interior, alternate exterior, and
corresponding angles are
congruent. They also conclude
that same-side interior and
same-side exterior angles
are supplementary. They also
conclude that the transversal
intersects two lines that are not
parallel, these relationships do
not hold true.
Problem 5
Measuring Angles Formed by Two Lines
and a Transversal
1. Draw a transversal intersecting two non-parallel lines, and number
each angle. Then use a protractor to determine each angle measure.
Answers will vary.
1 2
5 6
Use a
straightedge.
3 4
7 8
3
1
2
m/1 5 76°, m/6 5 76°, m/5 5 104°, m/2 5 104°,
m/3 5 115°, m/8 5 115°, m/4 5 65°, m/7 5 65°.
2. Draw a transversal intersecting two parallel lines, and number
each angle.Then use a protractor to determine each
angle measure.
Answers will vary.
Materials
1
5
2
6
3
Protractor
7
Straightedge
1
4
8
3
2
m/1 5 115°, m/6 5 115°, m/5 5 65°, m/2 5 65°,
Have students complete
Questions 1 through 9 with
a partner. Then share the
responses as a class.
m/3 5 115°, m/8 5 115°, m/4 5 65°, m/7 5 65°.
© 2011 Carnegie Learning
Grouping
Use the information from Questions 1 and 2 to answer Questions 3 through 8.
3. What do you notice about the measures of each pair of alternate interior angles when
the lines are:
a. non-parallel?
© 2011 Carnegie Learning
The alternate interior angles do not have equal measures.
Share Phase,
Questions 1 through 3
• What did you notice about
the measures of each pair of
vertical angles?
b. parallel?
The alternate interior angles have equal measures.
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 555
• Were any adjacent angles
equal in measure?
• Are there any perpendicular
lines in your drawing?
• Under what conditions are
the alternate interior angles
congruent?
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 555
Share Phase,
Questions 4 through 7
• Under what conditions are
the alternate exterior angles
congruent?
• Under what conditions are
the corresponding angles
congruent?
• Under what conditions are
the same-side interior angles
supplementary?
• Under what conditions are
the same-side exterior angles
supplementary?
4. What do you notice about the measures of each pair of alternate exterior angles when
the lines are:
a. non-parallel?
The alternate exterior angles do not have equal measures.
b. parallel?
The alternate exterior angles have equal measures.
5. What do you notice about the measures of each pair of corresponding angles when
the lines are:
a. non-parallel?
The corresponding angles do not have equal measures.
b. parallel?
The corresponding angles have equal measures.
6. What do you notice about the measures of the same-side interior angles when the
lines are:
a. non-parallel?
The same-side interior angles are not supplementary.
b. parallel?
7. What do you notice about the measures of the same-side exterior angles when the
lines are:
a. non-parallel?
The same-side exterior angles are not supplementary.
b. parallel?
© 2011 Carnegie Learning
The same-side interior angles are supplementary.
556 • Chapter 10 Line and Angle Relationships
556 • Chapter 10 Line and Angle Relationships
© 2011 Carnegie Learning
The same-side exterior angles are supplementary.
8. Summarize your conclusions in the table by writing the relationships of the measures
of the angles. The relationships are either congruent or not congruent, supplementary
or not supplementary.
Angles
Two Parallel
Lines Intersected
by a Transversal
Two Non-Parallel
Lines Intersected
by a Transversal
Alternate Interior Angles
congruent
not congruent
Alternate Exterior Angles
congruent
not congruent
Corresponding Angles
congruent
not congruent
Same-Side Interior Angles
supplementary
not supplementary
Same-Side Exterior Angles
supplementary
not supplementary
9. Use your table in Question 8 to compare your conclusions with other groups or
classmates. Also, compare the measures of the angles everyone used. What do
you notice?
Although we all had different measures for our angles, we had the same
© 2011 Carnegie Learning
© 2011 Carnegie Learning
conclusions in our tables.
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 557
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 557
Problem 6
Students solve a “Who’s
Correct?” problem which
focuses on corresponding
angles formed by two different
transversals intersecting
parallel lines. They will use their
knowledge of special pairs
of angles associated with a
transversal intersecting two
parallel lines to determine the
unknown measures of angles
in different situations. The third
situation can be expressed as
an equation and solved using
addition and division. The last
situation does require students
to extend a transversal such
that it intersects both parallel
lines forming congruent
alternate interior angles and a
triangle.
Problem 6
Solving for Unknown Angle Measures
Sylvia and Scott were working together to solve the problem shown.
___
___
Given: AB i CD . Solve for x. Show all your work.
E
xº
A
C
57º
57º
B
D
123 º
1. Sylvia concluded that x 5 66°. How did Sylvia get her answer?
Sylvia assumed that the corresponding angles on ray ED were congruent to the
angles formed on ray EC, so she solved for x by using the triangle at the top of the
figure: 180° 2 57° 2 57° 5 66°.
2. Scott does not agree with Sylvia’s answer. He thinks there is not enough information
to solve the problem. How could Scott alter the figure to explain his reason for
disagreeing with Sylvia’s answer?
Scott could redraw ray ED several different ways such that the measures of the
Have students complete
Questions 1 through 3 with
a partner. Then share the
responses as a class.
angles located at points B and D change. This would show Sylvia that the angles
formed on ray EC are not congruent to the angles formed on ray ED.
3. Who is correct?
Scott is correct. There is not enough information to solve for x.
Share Phase,
Questions 1 through 3
• How many transversals are in
© 2011 Carnegie Learning
Grouping
• Can you assume the lengths
of line segments EA and EB
are equal?
• If the length of line segment
EA is not equal to the length
of line segment EB, what
does this tell you about the
measure of ∠EAB and the
measure of ∠EBA?
558 • Chapter 10 Line and Angle Relationships
• Can you determine the measures of any angles formed at point B? Explain.
• Can you determine the measures of any angles formed at point D? Explain.
• Which angle measurements can you determine?
558 • Chapter 10 Line and Angle Relationships
© 2011 Carnegie Learning
this diagram?
Grouping
Have students complete
Questions 4 through 9 with
a partner. Then share the
responses as a class.
4. Opposite sides of this geometric figure are parallel. Suppose that the measure of
angle M is equal to 30°. Solve for the measures of angles G, E, and O. Explain
your reasoning.
G
Share Phase,
Questions 4 through 6
• In Question 4, what is the
relationship between ∠G
and ∠M?
E
M
O
The measure of angle G is equal to 150° because angle M and angle G are
same-side interior angles, so they are supplementary. The measure of angle E
is 30° because angle G and angle E are same-side interior angles, so they are
supplementary. The measure of angle O is 150° because angle E and angle O
are same-side interior angles, so they are supplementary.
• What is the measure of ∠G?
• What is the relationship
between ∠M and ∠O?
• What is the measure of ∠O?
• What is the relationship
5. Arrowheads indicate parallel lines. Determine the measures of all angles.
between ∠O and ∠E?
• What is the measure of ∠E?
• What is the relationship
34° 146°
34°
146°
34°
146°
34°
146°
between ∠E and ∠G?
• What is the measure of ∠G?
• In Question 5, which angles
6. Arrowheads indicate parallel lines. Determine the measures of all angles.
in the diagram are equal to
the measure of 34°?
angle adjacent to the
34° angle?
• In Question 6, how can the
angles labeled x and x1100
be used to write an equation
to solve for x?
© 2011 Carnegie Learning
140º
(x + 100)º
40º
140º
40º
© 2011 Carnegie Learning
• What is the measure of the
40º x º
140º
140º
40º
x 1 (x 1 100) 5 180
2x 5 80, so x 5 40
• What is the sum of the
measures of the two angles
labeled x and x1100?
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 559
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 559
Share Phase,
Questions 7 through
9
‹___›
‹___›
,
    ⊥ ​DE​
    
• In Question 7, if ​CE​
___
___›
___
___›
7. In this figure, AB i CD and CE  DE. Solve for x. Show all your work.
what is the measure of ∠E?
E
90°
• How can you determine the
132°
A
measure of ∠ACD?
42°
B
132° 48°
C
• How can you determine the
42°
42°
D
x°
measure of ∠BDC?
• In Question 8, will any
triangles in the diagram
help you to determine the
measure of an angle?
The value of x is 42.
8. Arrowheads indicate parallel lines, and boxes indicate that the angles are right angles.
Determine the measure of each angle in this figure.
55°
• What is the measure of the
44°
angle adjacent to the
46° angle?
• What does the little boxes in
44°
90°
the diagram represent?
61°
29°
105°
75°
75° 105°
• In Question 9, how can
you extend a transversal
so it forms a triangle in this
diagram?
• Is there a pair of alternate
134° 46°
46° 134°
134° 46°
46°
46°
134°
79°
105°
75°
75°
105°
29°
134° 46°
9. Solve for x.
interior angles you could use
to help solve this problem?
130°
x 5 116
560 • Chapter 10 Line and Angle Relationships
560 • Chapter 10 Line and Angle Relationships
© 2011 Carnegie Learning
64°
130° 50° 66°
© 2011 Carnegie Learning
xº
Talk the Talk
Students summarize the
situations in which special pairs
of angles are either congruent
or supplementary. They will
conclude that these situations
exist when a transversal has
intersected two parallel lines in
most cases.
Talk the Talk
If two lines are intersected by a transversal…
●
… when are alternate interior angles congruent?
When two parallel lines are intersected by a transversal, alternate interior angles
are congruent.
●
… when are alternate exterior angles congruent?
When two parallel lines are intersected by a transversal, alternate exterior angles
Grouping
Have students solve this
problem with a partner. Then
share the responses as a class.
are congruent.
●
… when are corresponding angles congruent?
When two parallel lines are intersected by a transversal, corresponding angles
are congruent.
Share Phase,
Talk the Talk
• When are alternate interior
●
… when are vertical angles congruent?
When two parallel or non-parallel lines are intersected by a transversal, vertical
angles not congruent?
angles are congruent.
• When are alternate exterior
angles not congruent?
●
• When are corresponding
… when are same-side interior angles supplementary?
When two parallel lines are intersected by a transversal, same-side interior angles
are supplementary.
angles not congruent?
angles not supplementary?
© 2011 Carnegie Learning
• When are same-side interior
●
• When are same-side exterior
angles not supplementary?
• When are adjacent angles
not supplementary?
• When are vertical angles not
© 2011 Carnegie Learning
congruent?
…when are same-side exterior angles supplementary?
When two parallel lines are intersected by a transversal, same-side exterior angles
are supplementary.
●
…when are adjacent angles supplementary?
When two parallel or non-parallel lines are intersected by a transversal, adjacent
angles are supplementary.
Be prepared to share your solutions and methods.
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 561
10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 561