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Unit 3 • Two-Dimensional Geometry and Similarity
151
Angle Pairs
Activity Focus
• Measuring angles • Angle pairsMaterials
• Protractors • Straightedges• Graphing utilities (optional)
Chunking the Activity
#1–2 #10–11 #17 #3–4 #12–14 #18–20
#5–7 #15 #21
#8–9 #16 #22–23
1
Think, Pair, Share Students have studied angle classifi cations in previous courses. This question serves as a review.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.© 2010 College Board. All rights reserved.
My Notes
Unit 3 • Two-Dimensional Geometry and Similarity 151 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Use Manipulatives
3.1
What’s Your Angle?
a. b.
c. d.
e.
Two rays with a common endpoint form an angle. Th e common endpoint is called the vertex. You can use a protractor to draw and measure angles in degrees.
1. Angles can be classifi ed by their measures. Name the three
classifi cations and then draw an example of each.
2. Measure each angle and classify it according to its measure.
READING MATH
To read this angle, say “angle ABC,” “angle CBA,”
or “angle B.” B A C ACADEMIC VOCABULARY angle
Acute angle: Right Angle: Obtuse Angle:
47°, acute 68°, acute 147°, obtuse 90°, right 110°, obtuse 151-158_SB_MS2_3-1_SE.indd 151 12/23/09 7:27:08 PM 151-158_SB_MS2_3-1_TE.indd 151 151-158_SB_MS2_3-1_TE.indd 151 2/9/10 6:36:17 PM2/9/10 6:36:17 PM
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152
SpringBoard® Mathematics with Meaning™ Level 23
Group Presentation Havestudents share their answers to this question on white boards.
4
Guess and Check Students are matching angles to form a complementary pair or a supplementary pair.Watch for students who conclude that each right angle in Parts b and f is comple-mentary to itself. Complecomple-mentary angles must be a pair of angles.
W c angleinP TEACHER TO TEACHER © 2010 College Board All rights reserved 151-158_SB_MS2_3-1_SE.indd 153 12/23/09 7:27:29 PM
© 2010 College Board. All rights reserved.
My Notes
152 SpringBoard®Mathematics with MeaningTM Level 2
SUGGESTED LEARNING STRATEGIES: Group Presentation, Guess and Check
Complementary angles are two angles whose sum is 90°. Supplementary angles are two angles whose sum is 180°.
Angles can be paired in many ways.
3. Compare and contrast complementary and supplementary
angles.
4. Name pairs of angles that form complementary or
supplementary angles. Justify you choices.
a. b. c. d. e. f. g. h. 37° 25° 31° 59° 53° 143°
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.
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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Unit 3 • Two-Dimensional Geometry and Similarity
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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.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 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.8
Work Backward9
Think/Pair/Share Students will not be able to give a complement because the 98° angle is obtuse.Suggested Assignment
CHECK YOUR UNDERSTANDING p. 158, #1–2
UNIT 3 PRACTICE p. 229, #1–3
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 fi nd the unknown angle.© 2010 College Board. All rights reserved.
Unit 3 • Two-Dimensional Geometry and Similarity 153
My Notes
5. Why are angles 1 and 2 in this diagram complementary?
1 2
6. Why are angles 1 and 2 in this diagram supplementary?
1 2
7. Can two obtuse angles be supplementary? Explain your
reasoning.
8. Give the complement and supplement, if possible, of an angle
that measures 32°.
9. Give the complement and supplement, if possible, of an angle
that measures 98°.
10. Two angles are complementary. One measures (2x)° and the
other measures 48°.
a. Write an equation to fi nd the measure of the missing angle.
b. Solve the equation for x.
c. What is the value of the other angle? Show your work.
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Group Presentation, Quickwrite, Work Backward, Create Representations
In Items 5 and 6, angles 1 and 2 are examples of adjacent angles. Adjacent angles have a common side but no common interior.
MATH TERMS
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.
Answers may vary. Sample answer: A straight angle (180°) is formed by ∠1 and ∠2, so ∠1 and ∠2 must be supplementary.
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.
Complement: 90° - 32° = 58°; Supplement: 180° - 32° = 148°
Complement: 90° - 98° = (-8°), so complement does not exist. Supplement: 180° - 98° = 82°. 2x + 48 = 90 x = 21 42°; 2(21) = 42 151-158_SB_MS2_3-1_SE.indd 153 12/23/09 7:27:29 PM © 2010 College Board . All rights reserved . 151-158_SB_MS2_3-1_SE.indd 152 12/23/09 7:27:21 PM 151-158_SB_MS2_3-1_TE.indd 153 151-158_SB_MS2_3-1_TE.indd 153 2/9/10 6:36:20 PM2/9/10 6:36:20 PM
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154
SpringBoard® Mathematics with Meaning™ Level 2a
Create Representations Thisquestion is similar to Item 10. Make sure that students substitute the value of x into the expression 3x to fi nd the measure of the unknown angle.
Differentiating Instruction
Challenge students who may be ready for a more complex problem after completing Item 11 with this problem:
∠ ABC and ∠ JKL are complementary.
m∠ABC =
(
__34x - 13
)
° and m∠ JKL = (3x - 17)°. Find themeasure of each angle. Show your work.
Answer: x = 32; m∠ABC = 11°,
m∠ JKL = 79°.
b
Use ManipulativesSome students may have an easier time recognizing vertical angles if they see them as two “v’s” that have a common vertex.
c
Think/Pair/Share Askstudents to share their conclusions on white boards for the entire class to see.
d
Quickwrite If necessary, remind students that the symbol for congruent is and that they should say that numbers are equal and fi gures are congruent.Have students study the fi gure below Item 14. You may want them to identify vertical angles and angles that form supplementary pairs before going on to the next question.
S a verticalan TEACHER TO TEACHER H fi You may w TEACHER TO TEACHER © 2010 College Board All rights reserved 151-158_SB_MS2_3-1_SE.indd 155 12/23/09 7:27:39 PM
© 2010 College Board. All rights reserved.
154 SpringBoard®Mathematics with MeaningTM Level 2 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 fi nd the measure of the missing angle.
b. Solve the equation for x.
c. What is the measure of the other angle? Show your work.
Vertical angles are formed when two lines intersect.
12. ∠1 and ∠3 are vertical angles. Find the measure of each angle
using your protractor.
1 3
2 4
13. Name another pair of vertical angles and fi nd the measure
of each angle.
14. What conjectures can you make about the measures of a pair
of vertical angles?
When two parallel lines are cut by a transversal, eight angles are formed. 1 2 3 4 5 6 7 8 Transversal It may be easier to measure the
angles if you extend the rays. Vertical angles share a vertex but have no common rays. MATH TERMS
3x + 123 = 180
x = 19
57°; 3(19) = 57
2 and 4; m∠2 = 50°, m∠4 = 50°
Vertical angles are congruent.
m∠1 = 130°; m∠3 = 130°
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Unit 3 • Two-Dimensional Geometry and Similarity
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e
Quickwrite (a, d), Use 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.Suggested Assignment
CHECK YOUR UNDERSTANDING p. 158, #3–5
UNIT 3 PRACTICE p. 229, #4–6
f
Group Presentation Help students to see that the interior angles are “inside” the parallel lines.For the students who need a concrete visual 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.
g
Think/Pair/Share Help students see that the exterior angles are outside the parallel lines.Differentiating Instruction
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.
F n representa TEACHER TO
TEACHER
© 2010 College Board. All rights reserved.
My Notes
Unit 3 • Two-Dimensional Geometry and Similarity 155 SUGGESTED LEARNING STRATEGIES: Quickwrite,
Group Presentation, Think/Pair/Share
Use the drawing of parallel lines cut by a transversal on page 154.
15. ∠4 and ∠8 are corresponding angles.
a. Why do you think that they are called corresponding angles?
b. Measure angles 4 and 8. m∠4 = m∠8 =
c. Name three more pairs of corresponding angles and give their measures.
d. What conjectures can you make about pairs of corresponding angles formed when parallel lines are cut by a transversal?
16. ∠3 and ∠6 are alternate interior angles.
a. Why are they called alternate interior angles?
b. Measure angles 3 and 6. m∠3 = m∠6 =
c. Name another pair of alternate interior angles and give their measures.
d. What conjectures can you make about pairs of alternate interior angles formed when parallel lines are cut by a transversal?
17. ∠2 and ∠7are alternate exterior angles.
a. Why are they called alternate exterior angles?
b. Measure angles 2 and 7. m∠2 = m∠7 = c. Name another pair of alternate exterior angles and give
their measures.
d. What can you conclude about pairs of alternate exterior angles formed when parallel lines are cut by a transversal?
68° 68°
Answers may vary. Sample answer: Corresponding angles are on the same side of the parallel lines and the transversal.
m∠1 m∠5 68°, m∠3 m∠7 112°, m∠2 m∠6 112°
They are congruent.
Answers may vary. Sample answer: Alternate interior angles are inside the parallel lines but on opposite sides of the transversal.
m∠4 = 68°, m∠5 = 68°
112° 112°
They are congruent.
m∠1 = m∠8 = 68°
They are congruent.
112° 112° Answers may vary. Sample answer: Alternate exterior angles are outside the parallel lines but on opposite sides of the transversal.
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156
SpringBoard® Mathematics with Meaning™ Level 2h
Look for a Pattern Thisquestion serves as a review of alternate interior angles.
i
Think/Pair/Share This question serves as a review of alternate exterior angles.j
Look for a Pattern This question gives students another opportunity to work with corresponding angles. © 2010 College Board All rights reserved 151-158_SB_MS2_3-1_SE.indd 157 12/23/09 7:27:47 PM© 2010 College Board. All rights reserved.
My Notes
156 SpringBoard®Mathematics with MeaningTM Level 2
18. In Figure A, two parallel lines are cut by a transversal. Th e
measure of ∠1 = 42°. Find m∠2 and describe the relationship that helped you determine the measure.
1 2 Figure A
19. In Figure B, two parallel lines are cut by a transversal. Th e
measure of ∠1 = 138°. Find m∠2 and describe the relationship that helped you determine the measure.
1
2 Figure B
20. In Figure C, two parallel lines are cut by a transversal. Th e
measure of ∠1 = 57°. Find m∠2 and describe the relationship that helped you determine the measure.
1 2 Figure C
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think/Pair/Share
Explanations may vary. Sample answer: m∠2 = 42°; alternate interior angles are congruent when two parallel lines are cut by a transversal.
Explanations may vary. Sample answer: m∠2 = 138°; alternate exterior angles are congruent when two parallel lines are cut by a transversal.
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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Unit 3 • Two-Dimensional Geometry and Similarity
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kl
Group Presentation Thesequestions serve as formative assessment of the activity. Encourage students to fi nd 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 classifi cation until all pairs have been identifi ed.
© 2010 College Board. All rights reserved.
My Notes
Unit 3 • Two-Dimensional Geometry and Similarity 157
Use the work you did and the conjectures you made in Questions 14–17 to help answer Questions 21 and 22.
21. In Figure D, lines j and k are parallel and are cut by
a transversal. Th e measure of ∠4 = 72°. Give the measures of the remaining seven angles. Justify your answers.
Angle Measure Explanation
1 2 3 4 72° Given. 5 6 7 8
22. Find an angle pair that supports your conjecture about each
type of angle pair.
a. Vertical angles
b. Corresponding angles
c. Alternate interior angles
d. Alternate exterior angles
SUGGESTED LEARNING STRATEGIES: Group Presentation
1 2 j k 4 3 5 6 7 8 Figure D
Answers may vary. Possible answers: angles 1 and 3, 2 and 4, 5 and 8, or 6 and 7
Answers may vary. Possible answers: angles 1 and 5, 4 and 7, 2 and 6, or 3 and 8
Answers may vary. Possible answers: angles 3 and 5 or 4 and 6
Answers may vary. Possible answers: angles 1 and 8 or 2 and 7
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. 108° ∠3 and ∠5 are alternate interior angles.
Alternate interior angles are congruent when lines are parallel.
72° ∠4 and ∠6 are alternate interior angles. Alternate interior angles are congruent when lines are parallel.
72° ∠1 and ∠7 are corresponding angles. Corresponding angles are congruent when lines are parallel.
108° ∠1 and ∠8 are alternate exterior angles. Alternate exterior angles are congruent when lines are parallel.
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158
SpringBoard® Mathematics with Meaning™ Level 2m
Visualize, Look for aPattern This question helps students see that their study of geometry does connect to their everyday lives.
Suggested Assignment
CHECK YOUR UNDERSTANDING p. 158, #6 UNIT 3 PRACTICE p. 229, #7 1a. complement: 58° supplement: 148˚ b. supplement: 67° 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° 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.
6. Answers may vary. Sample answer: students might draw
non-parallel lines cut by a transversal and measure the angles.
© 2010 College Board. All rights reserved.
158 SpringBoard®Mathematics with MeaningTM Level 2
SUGGESTED LEARNING STRATEGIES: Visualize, Look for a Pattern
23. Th ink 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.
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show your work.
1. Find the complement and/or supplement
of each angle.
a. 32° b. 113°
2. Find the measure of angle 1. Explain how
you found your answer.
a. b.
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.
1 2 3 73°
Two parallel lines are cut by a transversal as shown below.
1 8 2 7 3 6 4 5
5. Find each measure if m∠8 = 82°.
a. Find m∠3. Explain your answer.
b. Find m∠7. Explain your answer.
c. Find m∠5. Explain your answer.
6. MATHEMATICAL
R E F L E C T I O N
Suppose that a transversal 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 58°
1 137°
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.
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