Download What`s Your Angle? Angle Pairs

Angle Pairs
What’s Your Angle?
ACTIVITY
3.1
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Use Manipulatives
• Measuring angles
• Angle pairs
ACADEMIC VOCABULARY
angle
Materials
1. Angles can be classified by their measures. Name the three
classifications and then draw an example of each.
Acute angle:
Right Angle:
Obtuse Angle:
• Protractors
• Straightedges
• Graphing utilities (optional)
READING MATH
A
B
Chunking the Activity
C
2. Measure each angle and classify it according to its measure.
a. 47°, acute
Angle Pairs
Activity Focus
My Notes
Two rays with a common endpoint form an angle. The common
endpoint is called the vertex. You can use a protractor to draw and
measure angles in degrees.
ACTIVITY 3.1 Investigative
#1–2
#3–4
#5–7
#8–9
To read this angle, say
“angle ABC,” “angle CBA,”
or “angle B.”
b. 68°, acute
#10–11
#12–14
#15
#16
#17
#18–20
#21
#22–23
1 Think, Pair, Share Students
have studied angle classifications
in previous courses. This question
serves as a review.
© 2010 College Board. All rights reserved.
c. 147°, obtuse
d. 90°, right
2 Use Manipulatives Students
learned how to measure angles
with a protractor in earlier grades.
Remind them to line up the 0° line
with one side of the angle being
measured and to be sure that
the scale is oriented correctly. Do
not be too concerned if students
vary one or two degrees from the
measures given as answers.
e. 110°, obtuse
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151
ACTIVITY 3.1 Continued
ACTIVITY 3.1
continued
Angle Pairs
What’s Your Angle?
3 Group Presentation Have
students share their answers to
this question on white boards.
SUGGESTED LEARNING STRATEGIES: Group Presentation,
Guess and Check
My Notes
Angles can be paired in many ways.
4 Guess and Check Students
are matching angles to form
a complementary pair or a
supplementary pair.
W
Watch
for students who
c
conclude
that each right
P
angle in Parts
b and f is complementary to itself. Complementary
angles must be a pair of angles.
3. Compare and contrast complementary and supplementary
angles.
Complementary angles are
two angles whose sum is 90°.
Supplementary angles are two
angles whose sum is 180°.
TEACHER TO
TEACHER
Answers may vary. Sample answer: Complementary angles
have a sum of 90°, but supplementary angles have a sum of
180°. Two angles are needed to form both kinds of angle pairs.
Complementary angles will form a right angle if they are placed
next to each, but supplementary angles form a straight angle
when they are placed next to each other.
4. Name pairs of angles that form complementary or
supplementary angles. Justify you choices.
a.
b.
37°
25°
e.
d.
31°
© 2010 College Board. All rights reserved.
c.
f.
59°
g.
53°
h.
143°
The angles in Parts a and g are complementary because
37° + 53° = 90°; the angles in Parts b and f are
supplementary because 90° + 90° = 180°; the angles in
Parts d and e are complementary because 31° + 59° = 90°;
the angles in Parts a and h are supplementary because
37° + 143° = 180°.
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Angle Pairs
ACTIVITY 3.1
What’s Your Angle?
continued
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Group Presentation, Quickwrite, Work Backward,
Create Representations
5 Think/Pair/Share Make sure
students understand that these two
angles share a ray in such a way
that they form a 90º angle, thus
forming a complementary pair.
My Notes
5. Why are angles 1 and 2 in this diagram complementary?
MATH TERMS
1
In Items 5 and 6, angles 1 and
2 are examples of adjacent
angles. Adjacent angles have
a common side but no common
interior.
2
Answers may vary. Sample answer: ∠ABC is a right angle that
measures 90°. Since ∠ABC is formed by ∠1 and ∠2 , then ∠1
and ∠2 must be complementary.
draw two obtuse angles if they
need help understanding that
two obtuse angles cannot be
supplementary. There should be
several different counterexamples
within the class so that students
should clearly visualize this
concept.
2
Answers may vary. Sample answer: A straight angle (180°) is
formed by ∠1 and ∠2, so ∠1 and ∠2 must be supplementary.
7. Can two obtuse angles be supplementary? Explain your
reasoning.
Answers may vary. Sample answer: Obtuse angles have a measure
that is greater than 90°, so the sum of two obtuse angles would be
greater than 180°, so they could not be supplementary.
8 Work Backward
8. Give the complement and supplement, if possible, of an angle
that measures 32°.
9 Think/Pair/Share Students
will not be able to give a
complement because the 98°
angle is obtuse.
Complement: 90° - 32° = 58°; Supplement: 180° - 32° = 148°
© 2010 College Board. All rights reserved.
6 Group Presentation Students
should see that angles 1 and 2
form a straight line measuring 180°,
thus forming a supplementary pair.
7 Quickwrite Have students
6. Why are angles 1 and 2 in this diagram supplementary?
1
ACTIVITY 3.1 Continued
9. Give the complement and supplement, if possible, of an angle
that measures 98°.
Complement: 90° - 98° = (-8°), so complement does not exist.
Supplement: 180° - 98° = 82°.
Suggested Assignment
10. Two angles are complementary. One measures (2x)° and the
other measures 48°.
CHECK YOUR UNDERSTANDING
p. 158, #1–2
a. Write an equation to find the measure of the missing angle.
2x + 48 = 90
UNIT 3 PRACTICE
p. 229, #1–3
b. Solve the equation for x.
x = 21
c. What is the value of the other angle? Show your work.
42°; 2(21) = 42
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0 Create Representations This
question takes students to a more
abstract level. Be on the lookout
for students who think that the
measure of the second angle is 21°.
The value of x is 21. Students
must substitute the value of x
into the expression 2x to find the
unknown angle.
Unit 3 • Two-Dimensional Geometry and Similarity
153
ACTIVITY 3.1 Continued
a Create Representations This
question is similar to Item 10.
Make sure that students substitute
the value of x into the expression
3x to find the measure of the
unknown angle.
ACTIVITY 3.1
continued
Angle Pairs
What’s Your Angle?
My Notes
SUGGESTED LEARNING STRATEGIES: Create
Representations, Use Manipulatives, Think/Pair/Share,
Quickwrite
11. Two angles are supplementary. One angle measures (3x)° and
the other measures 123°.
a. Write an equation to find the measure of the missing angle.
3x + 123 = 180
Differentiating Instruction
∠ ABC and ∠ JKL are
complementary.
m∠ABC = _34_ x - 13 ° and
(
)
MATH TERMS
Vertical angles share a vertex
but have no common rays.
x = 19
c. What is the measure of the other angle? Show your work.
57°; 3(19) = 57
Vertical angles are formed when two lines intersect.
It may be easier to measure the
angles if you extend the rays.
12. ∠1 and ∠3 are vertical angles. Find the measure of each angle
using your protractor.
m∠ JKL = (3x - 17)°. Find the
1
2
measure of each angle. Show your
work.
3
4
m∠1 = 130°; m∠3 = 130°
Answer: x = 32; m∠ABC = 11°,
m∠ JKL = 79°.
13. Name another pair of vertical angles and find the measure
of each angle.
2 and 4; m∠2 = 50°, m∠4 = 50°
b Use Manipulatives
14. What conjectures can you make about the measures of a pair
of vertical angles?
S
Some
students may have
aan easier time recognizing
vertical an
angles if they see them as
two “v’s” that have a common
vertex.
TEACHER TO
TEACHER
© 2010 College Board. All rights reserved.
Challenge students who may be
ready for a more complex problem
after completing Item 11 with this
problem:
b. Solve the equation for x.
Vertical angles are congruent.
When two parallel lines are cut by a transversal, eight angles
are formed.
Transversal
1
c Think/Pair/Share Ask
2
3
students to share their conclusions
on white boards for the entire
class to see.
4
5
6
7
8
d Quickwrite If necessary,
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H
Have
students study the
figure below Item 14.
w
You may want
them to identify
vertical angles and angles that
form supplementary pairs before
going on to the next question.
TEACHER TO
TEACHER
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remind students that the symbol
for congruent is and that they
should say that numbers are equal
and figures are congruent.
Angle Pairs
ACTIVITY 3.1
What’s Your Angle?
continued
e Quickwrite (a, d), Use
SUGGESTED LEARNING STRATEGIES: Quickwrite,
Group Presentation, Think/Pair/Share
a. Why do you think that they are called corresponding angles?
Manipulatives (b), Think/Pair/
Share (c) Students may extend
the lines to make measuring the
angles easier. Ask students to
write the angle pairs using the
congruency symbol.
Answers may vary. Sample answer: Corresponding angles are
on the same side of the parallel lines and the transversal.
b. Measure angles 4 and 8.
m∠4 = 68° m∠8 = 68°
Suggested Assignment
My Notes
Use the drawing of parallel lines cut by a transversal on page 154.
15. ∠4 and ∠8 are corresponding angles.
CHECK YOUR UNDERSTANDING
p. 158, #3–5
c. Name three more pairs of corresponding angles and give
their measures.
m∠1 m∠5 68°, m∠3 m∠7 112°, m∠2 m∠6 112°
UNIT 3 PRACTICE
p. 229, #4–6
d. What conjectures can you make about pairs of corresponding
angles formed when parallel lines are cut by a transversal?
They are congruent.
f Group Presentation Help
16. ∠3 and ∠6 are alternate interior angles.
students to see that the interior
angles are “inside” the parallel
lines.
a. Why are they called alternate interior angles?
Answers may vary. Sample answer: Alternate interior angles are
inside the parallel lines but on opposite sides of the transversal.
b. Measure angles 3 and 6.
m∠3 = 112° m∠6 = 112°
FFor the students who
n
need a concrete visual
representa
representation of alternate interior
angles, you may want to use the
letter Z. Angles “tucked” inside
the letter Z are alternate interior
angles. You can show how to
change the orientation of the
letter Z so that it looks like the
letter N, or a backward Z, or a
backward N.
TEACHER TO
TEACHER
c. Name another pair of alternate interior angles and give their
measures.
© 2010 College Board. All rights reserved.
m∠4 = 68°, m∠5 = 68°
d. What conjectures can you make about pairs of alternate interior
angles formed when parallel lines are cut by a transversal?
They are congruent.
17. ∠2 and ∠7are alternate exterior angles.
a. Why are they called alternate exterior angles?
Answers may vary. Sample answer: Alternate exterior angles are
outside the parallel lines but on opposite sides of the transversal.
b. Measure angles 2 and 7.
m∠2 = 112° m∠7 = 112°
g Think/Pair/Share Help
students see that the exterior
angles are outside the parallel
lines.
c. Name another pair of alternate exterior angles and give
their measures.
m∠1 = m∠8 = 68°
d. What can you conclude about pairs of alternate exterior
angles formed when parallel lines are cut by a transversal?
Differentiating Instruction
They are congruent.
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© 2010 College Board. All rights reserved.
ACTIVITY 3.1 Continued
155
12/23/09 7:27:39 PM
Visual learners may have better
success at naming angle pairs if
they can represent the parallel
lines and transversal using pipe
cleaners and counters. Ask the
student to place 2 pipe cleaners
so that they are parallel to each
other, and then place another
pipe cleaner so that it intersects
both parallel lines. As you call out
an angle pair, such as “alternate
interior angles,” students will
identify 2 alternate interior angles
using their markers.
Unit 3 • Two-Dimensional Geometry and Similarity
155
ACTIVITY 3.1 Continued
h Look for a Pattern This
question serves as a review of
alternate interior angles.
ACTIVITY 3.1
continued
Angle Pairs
What’s Your Angle?
SUGGESTED LEARNING STRATEGIES: Look for a Pattern,
Think/Pair/Share
My Notes
18. In Figure A, two parallel lines are cut by a transversal. The
measure of ∠1 = 42°. Find m∠2 and describe the relationship
that helped you determine the measure.
i Think/Pair/Share This
question serves as a review of
alternate exterior angles.
Figure A
1
j Look for a Pattern This
2
question gives students another
opportunity to work with
corresponding angles.
Explanations may vary. Sample answer: m∠2 = 42°; alternate
interior angles are congruent when two parallel lines are cut by
a transversal.
19. In Figure B, two parallel lines are cut by a transversal. The
measure of ∠1 = 138°. Find m∠2 and describe the relationship
that helped you determine the measure.
Figure B
1
2
© 2010 College Board. All rights reserved.
Explanations may vary. Sample answer: m∠2 = 138°; alternate
exterior angles are congruent when two parallel lines are cut by
a transversal.
20. In Figure C, two parallel lines are cut by a transversal. The
measure of ∠1 = 57°. Find m∠2 and describe the relationship
that helped you determine the measure.
Figure C
2
1
Explanations may vary. Sample answer: m∠2 = 57°;
corresponding angles are congruent when two parallel lines
are cut by a transversal.
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Angle Pairs
ACTIVITY 3.1
What’s Your Angle?
continued
kl Group Presentation These
SUGGESTED LEARNING STRATEGIES: Group Presentation
My Notes
Use the work you did and the conjectures you made in
Questions 14–17 to help answer Questions 21 and 22.
Figure D
© 2010 College Board. All rights reserved.
21. In Figure D, lines j and k are parallel and are cut by
a transversal. The measure of ∠4 = 72°. Give the measures
of the remaining seven angles. Justify your answers.
Angle
1
2
Measure
3
4
5
108°
∠3 and ∠5 are alternate interior angles.
Alternate interior angles are congruent
when lines are parallel.
6
72°
∠4 and ∠6 are alternate interior angles.
Alternate interior angles are congruent
when lines are parallel.
7
72°
∠1 and ∠7 are corresponding angles.
Corresponding angles are congruent
when lines are parallel.
8
108°
∠1 and ∠8 are alternate exterior angles.
Alternate exterior angles are congruent
when lines are parallel.
1 2
4 3
5 6
7 8
j
k
Explanation
108°
∠1 and ∠4 are supplementary angles.
72°
∠2 and ∠4 are congruent because they
are vertical angles.
108°
∠1 and ∠3 are vertical angles.
72°
ACTIVITY 3.1 Continued
questions serve as formative
assessment of the activity.
Encourage students to find as
many different angle pairs as they
can for Parts a–d of Item 22. After
each presentation list the angle
pairs on the board under each
classification until all pairs have
been identified.
Given.
22. Find an angle pair that supports your conjecture about each
type of angle pair.
a. Vertical angles
Answers may vary. Possible answers: angles 1 and 3, 2 and
4, 5 and 8, or 6 and 7
b. Corresponding angles
Answers may vary. Possible answers: angles 1 and 5, 4 and
7, 2 and 6, or 3 and 8
c. Alternate interior angles
Answers may vary. Possible answers: angles 3 and 5 or
4 and 6
d. Alternate exterior angles
Answers may vary. Possible answers: angles 1 and 8 or
2 and 7
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© 2010 College Board. All rights reserved.
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Unit 3 • Two-Dimensional Geometry and Similarity
157
ACTIVITY 3.1 Continued
ACTIVITY 3.1
continued
Angle Pairs
What’s Your Angle?
m Visualize, Look for a
Pattern This question helps
students see that their study of
geometry does connect to their
everyday lives.
SUGGESTED LEARNING STRATEGIES: Visualize,
Look for a Pattern
23. Think about things, such as buildings, bridges, billboards, and
bicycles, which you see all around you. Describe examples of
angle pairs and angle relationships that you can observe in such
everyday objects.
Suggested Assignment
Answers may vary. Sample answer: The beams in the bridge
over the river form all kinds of angle pairs: vertical angles,
supplementary angles, corresponding angles, and alternate
interior and exterior angles.
CHECK YOUR UNDERSTANDING
p. 158, #6
UNIT 3 PRACTICE
p. 229, #7
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper.
Show your work.
2. Explanations may vary.
a. Sample answer: 32°; the
right angle symbol means
that the two angles must
be complementary, so the
measure of angle 1 is 32°.
b. Sample answer: 43°; a straight
line measures 180°, so the two
angles must be supplementary
making the measure of angle 1
equal to 43°.
3. 5x + 75 = 180
5x = 105
x = 21
m∠MNO = 105°
4. m∠1 = 73° because vertical
angles are congruent;
m∠2 = 107° because
angle 1 and angle 2 are
supplementary; m∠3 = 107°
because angles 2 and 3 are
vertical angles and vertical
angles are congruent.
5a. m∠3 = 98°
a. 32°
b. 113°
2. Find the measure of angle 1. Explain how
you found your answer.
a.
b.
1 8
2 7
3 6
4 5
5. Find each measure if m∠8 = 82°.
137°
1
1
a. Find m∠3. Explain your answer.
© 2010 College Board. All rights reserved.
b. supplement: 67°
1. Find the complement and/or supplement
of each angle.
b. Find m∠7. Explain your answer.
58°
3. ∠TUV and ∠MNO are supplementary.
m∠TUV = 75° and m∠MNO = (5x)°.
Find the measure of ∠MNO. Show
your work.
4. Find the measure of ∠1, ∠2, and ∠3.
Explain how you found each measure.
73°
2
3
c. Find m∠5. Explain your answer.
6. MATHEMATICAL Suppose that a transversal
R E F L E C T I O N intersects two lines that
are not parallel. Do you think that the angle
pairs that you studied in this activity would
still be congruent? Justify your answer.
1
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6. Answers may vary. Sample answer: students might draw
non-parallel lines cut by a transversal and measure the angles.
b. m∠7 = 98°
c. m∠5 = 98°
Explanations may vary.
Sample explanation for Part a:
m∠2 = 82° because ∠8
and ∠2 are vertical angles.
m∠3 = 98° because ∠3
and ∠2 are same-side
interior angles and are
supplementary.
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© 2010 College Board. All rights reserved.
1a. complement: 58°
supplement: 148˚
Two parallel lines are cut by a transversal as
shown below.