Download 19.1 Angles Formed by Intersecting Lines

LESSON
19.1
Name
Angles Formed by
Intersecting Lines
Class
Date
19.1 Angles Formed by Intersecting
Lines
Essential Question: How can you find the measures of angles formed by intersecting lines?
Resource
Locker
Common Core Math Standards
The student is expected to:
Explore 1
G-CO.9
Exploring Angle Pairs Formed by Intersecting Lines
When two lines intersect, like the blades of a pair of scissors, a number of angle pairs are formed.
You can find relationships between the measures of the angles in each pair.
Prove theorems about lines and angles.
Mathematical Practices
MP.3 Logic
Language Objective
Explain to a partner the differences among complementary angles,
supplementary angles, linear pairs, and vertical angles.
A
Possible answer:
ENGAGE
Possible answer: Identify any angles with known
angle measures. Look for pairs of vertical angles and
linear pairs of angles. Find angles that are
complementary (if any) and supplementary. Use
these relationships between pairs of angles
together with the known angle measures to find
the missing measures.
4
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©MNI/
Shutterstock
Essential Question: How can you find
the measures of angles formed by
intersecting lines?
Using a straightedge, draw a pair of intersecting lines like the open scissors. Label the angles
formed as 1, 2, 3, and 4.
B
3
2
Use a protractor to find each measure.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo,
making sure students understand that a multiplying
effect is created by the long handle and the short
“jaws.” Then preview the Lesson Performance Task.
1
Module 19
Angle
Measure of Angle
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120°
m∠2
60°
m∠3
120°
m∠4
60°
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180°
m∠3 + m∠4
180°
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180°
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Lesson 1
933
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IN1_MNLESE389762_U8M19L1 933
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HARDCOVER PAGES 753760
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Lesson 1
933
Module 19
9L1 933
62_U8M1
ESE3897
IN1_MNL
933
Lesson 19.1
4/19/14
12:15 PM
4/19/14 12:13 PM
You have been measuring vertical angles and linear pairs of angles. When two lines intersect, the angles that are
opposite each other are vertical angles. Recall that a linear pair is a pair of adjacent angles whose non-common sides
are opposite rays. So, when two lines intersect, the angles that are on the same side of a line form a linear pair.
EXPLORE 1
Reflect
1.
Exploring Angle Pairs Formed by
Intersecting Lines
Name a pair of vertical angles and a linear pair of angles in your diagram in Step A.
Possible answers: vertical angles: ∠1 and ∠3 or ∠2 and ∠4 ; linear pairs:
∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, or ∠1 and ∠4
INTEGRATE TECHNOLOGY
2.
Make a conjecture about the measures of a pair of vertical angles.
Vertical angles have the same measure and so are congruent.
3.
Use the Linear Pair Theorem to tell what you know about the measures of angles that form a linear pair.
The angles in a linear pair are supplementary, so the measures of the angles add
Students have the option of exploring the activity
either in the book or online.
up to 180°.
Explore 2
QUESTIONING STRATEGIES
Proving the Vertical Angles Theorem
How would you describe a pair of
supplementary angles in the drawing of
intersecting lines? adjacent angles
The conjecture from the Explore about vertical angles can be proven so it can be stated as a theorem.
The Vertical Angles Theorem
If two angles are vertical angles, then the angles are congruent.
4
1
3
Which pair of angles, if any, are congruent in a
drawing of two intersecting lines? Opposite
angles are congruent.
2
∠1 ≅ ∠3 and ∠2 ≅ ∠4
Follow the steps to write a Plan for Proof and a flow proof to prove the Vertical Angles
Theorem.
Given: ∠1 and ∠3 are vertical angles.
Prove: ∠1 ≅ ∠3
4
Module 19
1
3
2
934
EXPLORE 2
© Houghton Mifflin Harcourt Publishing Company
You have written proofs in two-column and paragraph proof formats. Another type of proof is called a flow proof. A
flow proof uses boxes and arrows to show the structure of the proof. The steps in a flow proof move from left to right
or from top to bottom, shown by the arrows connecting each box. The justification for each step is written below the
box. You can use a flow proof to prove the Vertical Angles Theorem.
Proving the Vertical Angles Theorem
QUESTIONING STRATEGIES
How do you identify vertical angles? They are
opposite angles formed by intersecting lines.
Lesson 1
PROFESSIONAL DEVELOPMENT
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Math Background
4/19/14 12:13 PM
Students will work with special angles. Congruent angles have the same measure.
A pair of angles is supplementary if their sum is 180°. A pair of angles is
complementary if their sum is 90°. Adjacent angles share one side without
overlapping. Vertical angles are the non-adjacent angles formed by intersecting
lines. They lie opposite each other at the point of intersection.
Angles Formed by Intersecting Lines 934
A
AVOID COMMON ERRORS
Complete the final steps of a Plan for Proof:
Because ∠1 and ∠2 are a linear pair and ∠2 and ∠3 are a linear pair, these pairs of angles
are supplementary. This means that m∠1 + m∠2 = 180° and m∠2 + m∠3 = 180°. By the
Transitive Property, m∠1 + m∠2 = m∠2 + m∠3. Next:
A common error that students make in writing
proofs is using the result they are trying to prove as
an intermediate step in the proof. Remind students to
be alert for this type of circular reasoning.
Subtract m∠2 from both sides to conclude that m∠1 = m∠3. So, ∠1 ≅ ∠3.
B
Use the Plan for Proof to complete the flow proof. Begin with what you know is true from
the Given or the diagram. Use arrows to show the path of the reasoning. Fill in the missing
statement or reason in each step.
CURRICULUM INTEGRATION
∠1 and ∠2 are a
linear pair.
The common error mentioned above is discussed in
the study of logic; it is known as the logical fallacy of
begging the question. This is not the same as the
general use of the phrase to “beg the question,” which
often means “to raise another question.”
Given
(see diagram)
∠1 and ∠2 are
supplementary .
Linear Pair
Theorem
m∠1 + m∠2 = 180°
Def. of
supplementary angles
∠1 and ∠3 are
vertical angles.
Given
∠2 and ∠3 are a
linear pair.
Given
(see diagram)
∠1 ≅ ∠3
© Houghton Mifflin Harcourt Publishing Company
Def. of
congruence
∠2 and ∠3 are
supplementary.
Linear Pair
Theorem
m∠2 + m∠3 = 180°
Def. of
supplementary angles
m∠1 = m∠3
m∠1 + m∠2 = m∠2 + m∠3
Subtraction
Property of
Equality
Transitive Property
of Equality
Reflect
4.
Discussion Using the other pair of angles in the diagram, ∠2 and ∠4, would a proof that ∠2 ≅ ∠4 also
show that the Vertical Angles Theorem is true? Explain why or why not.
Yes, it does not matter which pair of vertical angles is used in the proof. Similar statements
and reasons could be used for either pair of vertical angles.
5.
Draw two intersecting lines to form vertical angles. Label your lines and tell which angles are congruent.
Measure the angles to check that they are congruent.
A
D
E
C
Possible answer:
B
By the Vertical Angles Theorem, ∠AEC ≅ ∠DEB and ∠AED ≅ ∠CEB. Checking by
measuring, m∠AEC = m∠DEB = 45° and m∠AED = m∠CEB = 135°.
Module 19
935
Lesson 1
COLLABORATIVE LEARNING
IN1_MNLESE389762_U8M19L1 935
Small Group Activity
Have students review the different angle pairs and lesson illustrations. Have each
student draw one example of an angle pair, choosing from: a linear pair,
supplementary angles, complementary angles, and vertical angles. Students then
pass their drawings to others, who list all the information they know about that
type of angle pair. Have them keep passing the drawings for other students to add
more information until the drawing comes back to the student who drew it. Then
have students identify and draw the angle that is a complement of an acute angle,
a supplement of an obtuse angle, and the supplement of a right angle. acute,
acute, right
935
Lesson 19.1
4/19/14 12:13 PM
Explain 1
Using Vertical Angles
EXPLAIN 1
You can use the Vertical Angles Theorem to find missing angle measures in situations involving intersecting lines.
Example 1
Cross braces help keep the deck posts straight. Find the measure of
each angle.
Using Vertical Angles
QUESTIONING STRATEGIES
146°
5

6
If two lines intersect to form four angles and
one angle is an 80-degree angle, how do you
know there is another 80-degree angle? The
80-degree angle forms a linear pair with two
100-degree angles, and they each form a linear pair
with the remaining angle, so that angle must
measure 80 degrees.
7
∠6
Because vertical angles are congruent, m∠6 = 146°.

∠5 and ∠7
From Part A, m∠6 = 146°. Because ∠5 and ∠6 form a linear pair , they are
CONNECT VOCABULARY
supplementary and m∠5 = 180° - 146° = 34° . m∠
Differentiate the word vertical, meaning up and
down, and the idea of vertical angles, which share a
vertex but don’t need to be in an up-and-down
position. They are vertical in respect to one another.
also forms a linear pair with ∠6, or because it is a
7
= 34° because ∠ 7
vertical angle
with ∠5.
Your Turn
6.
The measures of two vertical angles are 58° and (3x + 4)°. Find the value of x.
58 = 3x + 4
18 = x
7.
The measures of two vertical angles are given by the expressions (x + 3)° and (2x - 7)°. Find the value
of x. What is the measure of each angle?
x + 3 = 2x - 7
x + 10 = 2x
10 = x
The measure of each angle is (x + 3)° = (10 + 3)° = 13°.
Module 19
936
© Houghton Mifflin Harcourt Publishing Company
54 = 3x
Lesson 1
DIFFERENTIATE INSTRUCTION
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Graphic Organizers
06/10/14 7:42 AM
Have students create a graphic organizer with a central oval titled Pairs of Angles
with leader lines to boxes for Adjacent angles, Linear pairs, Vertical angles,
Supplementary angles, and Complementary angles. Ask students to draw and
label an example of each pair of angles and to write a definition of each type
of angle pair.
Angles Formed by Intersecting Lines 936
Explain 2
EXPLAIN 2
Recall what you know about complementary and supplementary angles. Complementary angles are two angles
whose measures have a sum of 90°. Supplementary angles are two angles whose measures have a sum of 180°. You
have seen that two angles that form a linear pair are supplementary.
Using Supplementary and
Complementary Angles
Example 2
B
F
∠AFC and ∠CFD are a linear pair formed by an intersecting line and ray,
‹ ›
→
−
‾ , so they are supplementary and the sum of their measures is 180°. By the
AD and FC
diagram, m∠CFD = 90°, so m∠AFC = 180° - 90° = 90° and ∠AFC is also a right angle.
Because together they form the right angle ∠AFC, ∠AFB and ∠BFC are complementary and
the sum of their measures is 90°. So, m∠AFB = 90° - m∠BFC = 90° - 50° = 40°.
Find the measures of ∠DFE and ∠AFE.

What fact do you use to find the angle
measures in a linear pair? The angles in a
linear pair are supplementary.
solve for the measure of an angle in an angle-pair
relationship.Students will write an equation to
represent the angle-pair relationship and substitute
the information that they are given to solve for the
unknown.
CONNECT VOCABULARY
To help students remember the definitions of
complementary and supplementary, explain that the
letter c for complementary comes before s for
supplementary, just as 90° comes before 180°.
937
Lesson 19.1
E
Find the measures of ∠AFC and ∠AFB.

QUESTIONING STRATEGIES
∠BFA and ∠DFE are formed by two
so the angles are
vertical
intersecting lines
and are opposite each other,
angles. So, the angles are congruent. From Part A
m∠AFB = 40°, so m∠DFE = 40° also.
Because ∠BFA and ∠AFE form a linear pair, the angles are supplementary and the sum
© Houghton Mifflin Harcourt Publishing Company
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Encourage students to write an equation to
D
50°
A
supplementary angle might be. Encourage students to
state that the angle measure is close to, but not
exactly, 180°. Have them try to describe two lines that
intersect to form these angles.
How does an angle complementary to an
angle A compare with an angle supplementary
to A? It measures 90° less.
Use the diagram below to find the missing angle measures. Explain your
reasoning.
C
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Ask students what the maximum size of a
What fact do you use to find the angle
measures in a pair of complementary
angles? Two angles are complementary if the sum
of their measures is 90°.
Using Supplementary and Complementary Angles
of their measures is 180° . So, m∠AFE = 180° - m∠BFA = 180° - 40° = 140° .
Reflect
8.
In Part A, what do you notice about right angles ∠AFC and ∠CFD? Make a conjecture about right angles.
Possible answer: Both angles have measure 90°. Conjecture: All right angles are
congruent.
Module 19
937
Lesson 1
LANGUAGE SUPPORT
IN1_MNLESE389762_U8M19L1 937
4/19/14 12:13 PM
Visual Cues
Have students work in pairs to fill out an organizer like the following, starting
with prior knowledge:
Angle(s)
Right
Obtuse
Acute
Complementary
Supplementary
Definition
Picture
Your Turn
ELABORATE
You can represent the measures of an angle and its complement as x° and (90 - x)°.
Similarly, you can represent the measures of an angle and its supplement as x° and
(180 - x)°. Use these expressions to find the measures of the angles described.
9.
The measure of an angle is equal to
the measure of its complement.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 You may want to have students explore the
10. The measure of an angle is twice
the measure of its supplement.
x = 2(180 - x)
x = 90 - x
2x = 90
x = 360 - 2x
x = 45; so, 90 - x = 45
3x = 360
theorems in this lesson using geometry software. For
example, have students construct a linear pair of
angles, determine the measure of each angle, and use
the software’s calculate tool to find the sum of the
measures. Students can change the size of the angles
and see that the measures change while their sum
remains constant.
x = 120; so, 180 - x = 60
The measure of the angle is 45°
the measure of its complement is 45°.
The measure of the angle is 120°
the measure of its supplement is 60°.
Elaborate
11. Describe how proving a theorem is different than solving a problem and describe how they are the same.
Possible answer: A theorem is a general statement and can be used to justify steps in solving
problems, which are specific cases of algebraic and geometric relationships. Both proofs and
problem solving use a logical sequence of steps to justify a conclusion or to find a solution.
QUESTIONING STRATEGIES
12. Discussion The proof of the Vertical Angles Theorem in the lesson includes a Plan for Proof. How are a
Plan for Proof and the proof itself the same and how are they different?
Possible answer: A plan for a proof is less formal than a proof. Both start from the given
How many vertical angle pairs are formed
where three lines intersect at a point?
Explain. 6; if the small angles are numberered
consecutively 1–6, the vertical angles are 1 and 4, 2
and 5, 3 and 6, and the adjacent angle pairs 1–2 and
4–5, 2–3 and 5–6, 3–4, and 6–1.
information and reach the final conclusion, but a formal proof presents every logical step
in detail, while a plan describes only the key logical steps.
13. Draw two intersecting lines. Label points on the lines and tell what angles you
know are congruent and which are supplementary.
Possible answer:
K
L
J
H
Vertical angles are congruent: ∠GLK ≅ ∠JLH and ∠GLJ ≅ ∠KLH
Angles in a linear pair are supplementary: ∠GLJ and ∠GLK; ∠GLJ and ∠JLH; ∠JLH and
∠HLK; and ∠HLK and ∠GLK
© Houghton Mifflin Harcourt Publishing Company
G
SUMMARIZE THE LESSON
Have students make a graphic organizer to
show how some of the properties, postulates,
and theorems in this lesson build upon one another.
A sample is shown below.
Angle Addition Postulate
14. Essential Question Check-In If you know that the measure of one angle in a linear pair is 75°, how can
you find the measure of the other angle?
The angles in a linear pair are supplementary, so subtract the known measure from 180°:
Linear Pair Theorem
180 - 75 = 105, so the measure of the other angle is 105°.
Module 19
938
Lesson 1
Vertical Angles Theorem
IN1_MNLESE389762_U8M19L1 938
4/19/14 12:13 PM
Angles Formed by Intersecting Lines 938
EVALUATE
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
Use this diagram and information for Exercises 1–4.
Given: m∠AFB = m∠EFD = 50°
Points B, F, D and points E, F, C are collinear.
A
Concepts and Skills
Practice
Explore
Exploring Angle Pairs Formed by
Intersecting Lines
Exercise 1
Example 1
Proving the Vertical Angles Theorem
Exercises 2–11
1.
D
Determine whether each pair of angles is a pair of vertical angles, a linear pair of angles,
or neither. Select the correct answer for each lettered part.
Vertical
Linear Pair
A. ∠BFC and ∠DFE
B. ∠BFA and ∠DFE
C. ∠BFC and ∠CFD
D. ∠AFE and ∠AFC
E. ∠BFE and ∠CFD
F. ∠AFE and ∠BFC
Exercises 12–14
2.
Linear Pair
Neither
Vertical
Linear Pair
Neither
Vertical
Linear Pair
Neither
Vertical
Linear Pair
Neither
Vertical
Linear Pair
Neither
Find m∠AFE.
3.
m∠EFB = m∠AFB + m∠AFE = 80° + 50° = 130°
m∠AFE = 80°
4.
Find m∠DFC.
m∠DFC = m∠EFB, so m∠DFC = 130°
50° + m∠AFE + 50° = 180°
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Fresh
Picked/Alamy
Suggest still another way for students to remember
which sum is 90 and which is 180: complementary
and corner both start with the letter c, and corners are
often right angles. Supplementary and straight both
start with the letter s, and a straight line is a
180° angle.
Neither
Vertical
m∠AFB + m∠AFE + m∠EFD = 180°
CONNECT VOCABULARY
C
F
E
ASSIGNMENT GUIDE
Example 2
Using Supplementary and
Complementary Angles
B
Find m∠BFC.
m∠BFC = m∠EFD = 50°
5.
Represent Real-World Problems A sprinkler swings back and
forth between A and B in such a way that ∠1 ≅ ∠2, ∠1 and ∠3 are
complementary, and ∠2 and ∠4 are complementary. If m∠1 = 47.5°,
find m∠2, m∠3, and m∠4.
A
1
34
B
2
∠1 ≅ ∠2, so m∠2 = 47.5°
∠1 and ∠3 are complementary, so m∠3 = 90 - 47.5 = 42.5°
∠2 and ∠4 are complementary, so m∠4 = 90 - 47.5 =42.5°
Module 19
Exercise
IN1_MNLESE389762_U8M19L1 939
939
Lesson 19.1
Lesson 1
939
Depth of Knowledge (D.O.K.)
Mathematical Practices
1
1 Recall of Information
MP.4 Modeling
2–4
1 Recall of Information
MP.2 Reasoning
5–7
2 Skills/Concepts
MP.2 Reasoning
8–11
1 Recall of Information
MP.3 Logic
12–14
2 Skills/Concepts
MP.3 Logic
15
2 Skills/Concepts
MP.4 Modeling
9/23/14 7:50 AM
Determine whether each statement is true or false. If false, explain why.
6.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 As students answer the questions using the
If an angle is acute, then the measure of its complement must be greater than the measure of its
supplement.
False. The measure of an acute angle is less than 90°, so the measure of its complement
will be less than 90° and the measure of its supplement will be greater than 90°. So, the
measure of the supplement will be greater than the measure of the complement.
7.
angle names in a pair of angles, encourage them to
trace each angle along its letters and say the angle’s
name aloud as they write it. This will help students
get used to naming an angle with three letters.
A pair of vertical angles may also form a linear pair.
False. Vertical angles do not share a common side.
8.
If two angles are supplementary and congruent, the measure of each angle is 90°.
9.
If a ray divides an angle into two complementary angles, then the original angle is a right angle.
True
AVOID COMMON ERRORS
True
You can represent the measures of an angle and its complement as x° and (90 - x)°.
Similarly, you can represent the measures of an angle and its supplement as x° and
(180 - x)°. Use these expressions to find the measures of the angles described.
Some students may have difficulty organizing the
reasons or steps for a proof. You may wish to have
students write the given reasons or steps on sticky
notes. Then they can rearrange the sticky notes as
needed to write the proof in the correct order.
10. The measure of an angle is three times the measure of its supplement.
x = 3(180 - x)
x = 540 - 3x
4x = 540
x = 135; so, 180 - x = 45
The measure of the angle is 135° and the measure of its supplement is 45°.
11. The measure of the supplement of an angle is three times the measure of its complement.
180 - x = 3(90 - x)
180 - x = 270 - 3x
2x = 90
© Houghton Mifflin Harcourt Publishing Company
x = 45; so, 90 - x = 45 and 180 - x = 135
The measure of the angle is 45°, the measure of its complement is 45°, and the
measure of its supplement is 135°.
12. The measure of an angle increased by 20° is equal to the measure of its complement.
x + 20 = 90 - x
2x = 70
x = 35; so, 90 - x = 55
The measure of the angle is 35°, the measure of its complement is 55°.
Module 19
Exercise
IN1_MNLESE389762_U8M19L1 940
Lesson 1
940
Depth of Knowledge (D.O.K.)
Mathematical Practices
16–17
3 Strategic Thinking
MP.2 Reasoning
18–20
3 Strategic Thinking
MP.3 Logic
4/19/14 12:12 PM
Angles Formed by Intersecting Lines 940
Write a plan for a proof for each theorem.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Remind students that the shape of the letter X
13. If two angles are congruent, then their complements are congruent.
Given: ∠ABC ≅ ∠DEF
Prove: The complement of ∠ABC ≅ the complement of ∠DEF .
Plan for Proof:
If ∠ABC ≅ ∠DEF, then m∠ABC = m∠DEF.
suggests the idea of vertical angles.
The measure of the complement of ∠ABC = 90° - m∠ABC.
The measure of the complement of ∠DEF = 90° - m∠DEF.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Remind students of the definitions of vertical,
Since m∠ABC = m∠DEF, the measure of the complement of ∠DEF = 90° - m∠ABC.
Therefore, the measure of the complement of ∠ABC = the measure of the complements of
∠DEF.
The measures of the complements of the angles are equal, so the complements of the angles
are congruent.
complementary, supplementary, and linear pairs.
Have students use mental math to find complements
and supplements of angles before using variables
or expressions.
14. If two angles are congruent, then their supplements are congruent.
Given: ∠ABC ≅ ∠DEF
Prove: The supplement of ∠ABC ≅ the supplement of ∠DEF .
Plan for Proof:
If ∠ABC ≅ ∠DEF, then m∠ABC ≅ m∠DEF.
The measure of the supplement of ∠ABC = 180° - m∠ABC.
The measure of the supplement of ∠DEF = 180° - m∠DEF.
Since m∠ABC = m∠DEF, the measure of the supplement of ∠DEF = 180° - m∠ABC.
© Houghton Mifflin Harcourt Publishing Company
Therefore, the measure of the supplement of ∠ABC = the measure of the supplement of
∠DEF.
The measures of the supplements of the angles are equal, so the supplements of the angles
are congruent.
Module 19
IN1_MNLESE389762_U8M19L1 941
941
Lesson 19.1
941
Lesson 1
9/23/14 7:53 AM
15. Justify Reasoning Complete the two-column proof for the theorem “If two angles are congruent, then
their supplements are congruent.”
Statements
Reasons
1. ∠ABC ≅ ∠DEF
1. Given
2. The measure of the supplement of
∠ABC = 180° - m∠ABC.
2. Definition of the
3. The measure of the supplement of
∠DEF = 180° - m∠DEF .
3. Definition of the supplement of an angle
4.
m∠ABC = m∠DEF
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 If students have difficulty supplying the
missing reasons in a proof, give them the reasons in
random order and ask them to write the correct
reason in the appropriate line of the proof. This type
of scaffolding can be helpful for English language
learners until they become more comfortable with
the vocabulary of deductive reasoning.
supplement
of an angle
4. If two angles are congruent, their measures are
equal.
5. The measure of the supplement of
∠DEF = 180° - m∠ABC.
5. Substitution Property of
6. The measure of the supplement of ∠ABC
= the measure of the supplement of
∠DEF.
6.
7. The supplement of ∠ABC ≅ the
supplement of ∠DEF .
7. If the measures of the supplements of two angles
are equal, then supplements of the angles are
congruent.
Equality
Substitution Property of Equality
16. Probability The probability P of choosing an object at random from a group of objects is found by the
Number of favorable outcomes . Suppose the angle measures 30°, 60°, 120°, and 150°
fraction P(event) = ___
Total number of outcomes
are written on slips of paper. You choose two slips of paper at random.
© Houghton Mifflin Harcourt Publishing Company
a. What is the probability that the measures you choose are complementary?
1 possible pair (30°, 60°)
1
P(complementary) = ___ = _
6
6 angle pairs total
b. What is the probability that the measures you choose are supplementary?
2 possible pairs (30°, 150° and 60°, 120°)
2 =_
1
P(supplementary) = ____ = _
6 3
6 angle pairs total
Module 19
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942
Lesson 1
4/19/14 12:12 PM
Angles Formed by Intersecting Lines 942
PEER-TO-PEER DISCUSSION
H.O.T. Focus on Higher Order Thinking
Ask students to discuss with a partner the names and
diagrams for various pairs of angles introduced in
this lesson. Ask them to write and solve a word
problem using complementary or supplementary
angles on index cards, then switch cards with their
partners to solve the problem.
17. Communicate Mathematical Ideas Write a proof of the Vertical Angles Theorem in paragraph
proof form.
1
4
3
2
Given: ∠2 and ∠4 are vertical angles.
Prove: ∠2 ≅ ∠4
In the diagram of intersecting lines, ∠2 and ∠4 are vertical angles. Also, ∠2 and ∠3 are
a linear pair and ∠3 and ∠4 are a linear pair. By the Linear Pair Theorem, ∠2 and ∠3 are
supplementary and ∠3 and ∠4 are supplementary. Then m∠2 + m∠3 = 180° and
m∠3 + m∠4 = 180° by the definition of supplementary angles. By the Transitive Property
of Equality, m∠2 + m∠3 = m∠3 + m∠4. Using the Subtraction Property of Equality,
m∠2 = m∠4. So, ∠2 ≅ ∠4 by the definition of congruence.
JOURNAL
Have students write about each angle-pair
relationship discussed in this lesson (supplementary
angles, complementary angles, and vertical angles).
Students should include definitions and drawings to
support their descriptions.
18. Analyze Relationships If one angle of a linear pair is acute, then the other angle must be obtuse.
Explain why.
The sum of the measures of the two angles of a linear pair must be 180°. If the measure of
one angle is less than 90°, then the measure of the other angle must be greater than 90°.
19. Critique Reasoning Your friend says that there is an angle whose measure is the same as the measure
of the sum of its supplement and its complement. Is your friend correct? What is the measure of the angle?
Explain your friend’s reasoning.
Yes; 90°: the measure of its complement is 0°, and the measure of its supplement is 90°, so
0° + 90° = 90°.
20. Critical Thinking Two statements in a proof are:
© Houghton Mifflin Harcourt Publishing Company
m∠A = m∠B
m∠B = m∠C
What reason could you give for the statement m∠A = m∠C? Explain your reasoning.
You could use either the Transitive Property of Equality, since both statements have m∠B
in common; or you could use the Substitution Property of Equality by substituting m∠C for
m∠B in the first statement.
Module 19
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943
Lesson 19.1
943
Lesson 1
4/19/14 12:12 PM
Lesson Performance Task
CONNECT VOCABULARY
The image shows the angles formed by a pair of scissors. When the
scissors are closed, m∠1 = 0°. As the scissors are opened, the measures
of all four angles change in relation to each other. Describe how the
measures change as m∠1 increases from 0° to 180°.
∠3
∠4
∠2
A supplement is something that completes something
else, as a vitamin supplement may complete a person’s
daily vitamin requirement. The supplement of an
angle completes a linear pair, with measures that
complete a sum of 180°. A complement similarly
brings something to completion, like a new coach
who proves to be a perfect complement to the team.
The complement of an angle brings a right angle to
completion.
∠1
When m∠1 = 0°, m∠3 = 0°, and the measures of both ∠2 and ∠4 equal 180°.
As m∠1 increases, the measure of ∠3 increases as well, so that m∠3 always
equals m∠1. At the same time, the measures of ∠2 and ∠4 decrease steadily,
both angles being congruent to each other and supplementary to ∠1.
When m∠1 = 90°, the measures of the other three angles are also 90°.
AVOID COMMON ERRORS
When m∠1 > 90°, the measures of ∠2 and ∠4 are less than 90° and continually
Students may have difficulty with the concept that as
angle measures increase from 0° to 180°, their
supplements change from obtuse to right to acute
angles. Stress that supplementary angles have
measures whose sum is 180°, and that one angle in a
pair of supplementary angles that are not right angles
will always be obtuse and one will be acute, and that
supplements of right angles are right angles.
decrease, reaching 0° when m∠1 = 180°.
© Houghton Mifflin Harcourt Publishing Company
Module 19
944
Lesson 1
EXTENSION ACTIVITY
IN1_MNLESE389762_U8M19L1 944
A lever is a simple machine that enables the user to lift or move an object using
less force than the weight of the object. A teeter-totter is an example of a “class 1”
lever, which has a fulcrum between the object being moved and the applied force.
Have students research “double class 1 levers,” of
Class 1 Lever
which pliers and scissors are examples. Students
Force
Load
should explain how this type of lever differs from
single class 1 levers, and describe how double class
1 levers create a mechanical advantage that allows
Fulcrum
users to produce forces greater than they apply.
4/19/14 12:12 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Angles Formed by Intersecting Lines 944