4.1 Angles Formed by Intersecting Lines - Lake County Angles Formed by Intersecting Lines Essential...

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Explore 1 Exploring Angle Pairs Formed by Intersecting LinesWhen 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.

A Using a straightedge, draw a pair of intersecting lines like the open scissors. Label the angles formed as 1, 2, 3, and 4.

B Use a protractor to find each measure.

Resource Locker

14 2

3

Angle Measure of Angle

m∠1

m∠2

m∠3

m∠4

m∠1 + m∠2

m∠2 + m∠3

m∠3 + m∠4

m∠1 + m∠4

120°

Possible answer:

60°

120°

60°

180°

180°

180°

180°

Module 4 163 Lesson 1

4.1 Angles Formed by Intersecting Lines

Essential Question: How can you find the measures of angles formed by intersecting lines?

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GE_MNLESE385795_U2M04L1.indd 163 01/04/14 11:37 PM

Common Core Math StandardsThe student is expected to:

G-CO.C.9

Prove theorems about lines and angles.

Mathematical Practices

MP.3 Logic

Language ObjectiveExplain to a partner the differences among complementary angles, supplementary angles, linear pairs, and vertical angles.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 145152

Turn to these pages to find this lesson in the hardcover student edition.

Angles Formed by Intersecting Lines

ENGAGE Essential Question: How can you find the measures of angles formed by intersecting lines?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.

PREVIEW: LESSON PERFORMANCE TASKView 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.

163

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore 1 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.

Using a straightedge, draw a pair of intersecting lines like the open scissors. Label the angles

formed as 1, 2, 3, and 4.

Use a protractor to find each measure.

Resource

Locker

G-CO.C.9 Prove theorems about lines and angles.

COMMONCORE

14 2

3

AngleMeasure of Angle

m∠1

m∠2

m∠3

m∠4

m∠1 + m∠2

m∠2 + m∠3

m∠3 + m∠4

m∠1 + m∠4

120°

Possible answer:

60°

120°

60°

180°

180°

180°

180°

Module 4

163

Lesson 1

4 . 1 Angles Formed by Intersecting

Lines

Essential Question: How can you find the measures of angles formed by intersecting lines?

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01/04/14 11:38 PM

163 Lesson 4 . 1

L E S S O N 4 . 1

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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.

Reflect

1. Name a pair of vertical angles and a linear pair of angles in your diagram in Step A.

2. Make a conjecture about the measures of a pair of vertical angles.

3. Use the Linear Pair Theorem to tell what you know about the measures of angles that form a linear pair.

Explore 2 Proving the Vertical Angles TheoremThe 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.

∠1 ≅ ∠3 and ∠2 ≅ ∠4

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.

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

14 2

3

1

324

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

Vertical angles have the same measure and so are congruent.

The angles in a linear pair are supplementary, so the measures of the angles add

up to 180°.




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Math BackgroundStudents 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.

EXPLORE 1 Exploring Angle Pairs Formed by Intersecting Lines

INTEGRATE TECHNOLOGYStudents have the option of exploring the activity either in the book or online.

QUESTIONING STRATEGIESHow would you describe a pair of supplementary angles in the drawing of

intersecting lines? adjacent angles

Which pair of angles, if any, are congruent in a drawing of two intersecting lines? Opposite

angles are congruent.

EXPLORE 2 Proving the Vertical Angles Theorem

QUESTIONING STRATEGIESHow do you identify vertical angles? They are

opposite angles formed by intersecting lines.

PROFESSIONAL DEVELOPMENT

Angles Formed by Intersecting Lines 164


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A 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:

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.

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.

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.

E

D

B

A

C

Given(see diagram)

Given

∠1 and ∠3 arevertical angles.

∠1 and ∠2 are alinear pair.

Linear PairTheorem

Def. ofsupplementary angles

Def. ofcongruence

Linear PairTheorem

SubtractionProperty ofEquality

Def. of

supplementary angles

Given(see diagram)

∠1 and ∠2 are

supplementary

m∠1 + m∠2 = m∠2 + m∠3m∠1 = m∠3

Property

of Equality

Transitive

m∠1 + m∠2 = 180°

∠2 and ∠3 aresupplementary.

∠2 and are alinear pair.

∠3 + = 180°m∠2 m∠3

∠1 ∠3≅

.

Subtract m∠2 from both sides to conclude that m∠1 = m∠3. So, ∠1 ≅ ∠3.

By the Vertical Angles Theorem, ∠AEC ≅ ∠DEB and ∠AED ≅ ∠CEB . Checking by

measuring, m∠AEC = m∠DEB = 45° and m∠AED = m∠CEB = 135°.

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.

Possible answer:




COLLABORATIVE LEARNING

Small Group ActivityHave 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

AVOID COMMON ERRORSA 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.

CURRICULUM INTEGRATIONThe 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.”

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Explain 1 Using Vertical AnglesYou 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.

∠6

Because vertical angles are congruent, m∠6 = 146°.

∠5 and ∠7

From Part A, m∠6 = 146°. Because ∠5 and ∠6 form a , they are

supplementary and m∠5 = 180° - 146° = . m∠ = because ∠

also forms a linear pair with ∠6, or because it is a with ∠5.

Your Turn

6. The measures of two vertical angles are 58° and (3x + 4) °. Find the value of 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?

675

146°

linear pair

vertical angle

58 = 3x + 4

54 = 3x

18 = x

x + 3 = 2x - 7

x + 10 = 2x

10 = x

The measure of each angle is (x + 3) ° = (10 + 3) ° = 13°.

34° 34°7 7


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DIFFERENTIATE INSTRUCTION

Graphic OrganizersHave 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.

EXPLAIN 1 Using Vertical Angles

QUESTIONING STRATEGIESIf 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.

CONNECT VOCABULARY 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.



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Explain 2 Using Supplementary and Complementary AnglesRecall 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.

Example 2 Use the diagram below to find the missing angle measures. Explain your reasoning.

Find the measures of ∠AFC and ∠AFB.

∠AFC and ∠CFD are a linear pair formed by an intersecting line and ray,

‹ − › AD and

‾→ FC , so they are supplementary and the sum of their measures is 180°. By the

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.

∠BFA and ∠DFE are formed by two and are opposite each other,

so the angles are angles. So, the angles are congruent. From Part A

m∠AFB = 40°, so m∠DFE = also.

Because ∠BFA and ∠AFE form a linear pair, the angles are and the sum

of their measures is . So, m∠AFE = - m∠BFA = - = .

Reflect

8. In Part A, what do you notice about right angles ∠AFC and ∠CFD? Make a conjecture about right angles.

F

A

B

C

D

E

50°

intersecting lines

vertical

supplementary

40°

180° 180° 40° 140°180°

Possible answer: Both angles have measure 90°. Conjecture: All right angles are

congruent.


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LANGUAGE SUPPORT

Visual CuesHave students work in pairs to fill out an organizer like the following, starting with prior knowledge:

Angle(s) Definition PictureRight

Obtuse

Acute

Complementary

Supplementary

EXPLAIN 2 Using Supplementary and Complementary Angles

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Ask students what the maximum size of 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.

QUESTIONING STRATEGIESWhat fact do you use to find the angle measures in a linear pair? The angles in a

linear pair are supplementary.

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°.

How does an angle complementary to an angle A compare with an angle supplementary

to A? It measures 90° less.

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Encourage students to write an equation to 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°.

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Your Turn

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.

10. The measure of an angle is twice the measure of its supplement.

Elaborate

11. Describe how proving a theorem is different than solving a problem and describe how they are the same.

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?

13. Draw two intersecting lines. Label points on the lines and tell what angles you know are congruent and which are supplementary.

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?

L

KG

HJ

x = 90 - x

2x = 90

x = 45; so, 90 - x = 45

The measure of the angle is 45° the measure of its complement is 45°.

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.

x = 2 (180 - x)

x = 360 - 2x

3x = 360

x = 120; so, 180 - x = 60

The measure of the angle is 120° the measure of its supplement is 60°.

Possible answer: A plan for a proof is less formal than a proof. Both start from the given

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.

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

The angles in a linear pair are supplementary, so subtract the known measure from 180°:

180 - 75 = 105, so the measure of the other angle is 105°.

Possible answer:





Angle Addition Postulate

Linear Pair Theorem

Vertical Angles Theorem

ELABORATE INTEGRATE MATHEMATICAL PRACTICESFocus on TechnologyMP.5 You may want to have students explore the 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.

QUESTIONING STRATEGIESHow 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.

SUMMARIZE THE LESSONHave 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.



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• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and 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.

1. 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.A. ∠BFC and ∠DFE Vertical Linear Pair NeitherB. ∠BFA and ∠DFE Vertical Linear Pair NeitherC. ∠BFC and ∠CFD Vertical Linear Pair NeitherD. ∠AFE and ∠AFC Vertical Linear Pair NeitherE. ∠BFE and ∠CFD Vertical Linear Pair NeitherF. ∠AFE and ∠BFC Vertical Linear Pair Neither

2. Find m∠AFE. 3. Find m∠DFC.

4. Find m∠BFC.

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

E

B C

DF

BA43

1 2

m∠AFB + m∠AFE + m∠EFD = 180°

50° + m∠AFE + 50° = 180°

m∠AFE = 80°

m∠EFB = m∠AFB + m∠AFE = 80° + 50° = 130°

m∠DFC = m∠EFB, so m∠DFC = 130°

m∠BFC = m∠EFD = 50°

∠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°



GE_MNLESE385795_U2M04L1 169 22/05/14 12:59 PMExercise Depth of Knowledge (D.O.K.)COMMONCORE 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

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreExploring Angle Pairs Formed by Intersecting Lines

Exercise 1

Example 1Using Vertical Angles

Exercises 2–11

Example 2Using Supplementary and Complementary Angles

Exercises 12–14

CONNECT VOCABULARY 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.

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Determine whether each statement is true or false. If false, explain why.

6. If an angle is acute, then the measure of its complement must be greater than the measure of its supplement.

7. A pair of vertical angles may also form a linear pair.

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.

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.

10. The measure of an angle is three times the measure of its supplement.

11. The measure of the supplement of an angle is three times the measure of its complement.

12. The measure of an angle increased by 20° is equal to the measure of its complement.

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.

False. Vertical angles do not share a common side.

True

True

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°.

180 - x = 3 (90 - x)

180 - x = 270 - 3x

2x = 90

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°.

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°.




GE_MNLESE385795_U2M04L1.indd 170 20/03/14 8:13 AMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

16–17 3 Strategic Thinking MP.2 Reasoning

18–20 3 Strategic Thinking MP.3 Logic

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 As students answer the questions using the 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.

AVOID COMMON ERRORSSome 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.



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Write a plan for a proof for each theorem.

13. If two angles are congruent, then their complements are congruent.

Given: ∠ABC ≅ ∠DEF

Prove: The complement of ∠ABC ≅ the complement of ∠DEF .

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 complement of ∠ABC = 90° - m∠ABC.

The measure of the complement of ∠DEF = 90° - m∠DEF.

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.

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.

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.


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GE_MNLESE385795_U2M04L1 171 10/14/14 6:57 PM

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Remind students that the shape of the letter X suggests the idea of vertical angles.

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Remind students of the definitions of vertical, complementary, supplementary, and linear pairs. Have students use mental math to find complements and supplements of angles before using variables or expressions.

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15. Justify Reasoning Complete the two-column proof for the theorem “If two angles are congruent, then their supplements are congruent.”

16. Probability The probability P of choosing an object at random from a group of objects is found by the

fraction P(event) = Number of favorable outcomes ___ Total number of outcomes

. Suppose the angle measures 30°, 60°, 120°, and 150°

are written on slips of paper. You choose two slips of paper at random.

a. What is the probability that the measures you choose are complementary?

b. What is the probability that the measures you choose are supplementary?

Statements Reasons

1. ∠ABC ≅ ∠DEF 1. Given

2. The measure of the supplement of ∠ABC = 180° - m∠ABC.

2. Definition of the of an angle

3. The measure of the supplement of ∠DEF = 180° - m∠DEF . 3.

4. 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 .

7. If the measures of the supplements of two angles are equal, then supplements of the angles are congruent.

m∠ABC = m∠DEF

∠DEF

supplement

Equality

Definition of the supplement of an angle

Substitution Property of Equality

P(complementary) = 1 possible pair (30°, 60°)

___ 6 angle pairs total

= 1 _ 6

P(supplementary) = 2 possible pairs (30°, 150° and 60°, 120°)

____ 6 angle pairs total

= 2 _ 6

= 1 _ 3





INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.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.



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H.O.T. Focus on Higher Order Thinking

17. Communicate Mathematical Ideas Write a proof of the Vertical Angles Theorem in paragraph proof form.

Given: ∠2 and ∠4 are vertical angles.

Prove: ∠2 ≅ ∠4

18. Analyze Relationships If one angle of a linear pair is acute, then the other angle must be obtuse. Explain why.

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.

20. Critical Thinking Two statements in a proof are:

m∠A = m∠B

m∠B = m∠C

What reason could you give for the statement m∠A = m∠C? Explain your reasoning.

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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.

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°.

Yes; 90°: the measure of its complement is 0°, and the measure of its supplement is 90°, so 0° + 90° = 90°.

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.




PEERTOPEER DISCUSSIONAsk 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.

JOURNALHave 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.

173 Lesson 4 . 1


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Lesson Performance Task

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°. ∠4

∠2∠1∠3

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°.

When m∠1 > 90°, the measures of ∠2 and ∠4 are less than 90° and continually

decrease, reaching 0° when m∠1 = 180°.


DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-A;CA-A


Class 1 LeverLoad Force

Fulcrum

EXTENSION ACTIVITY

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 which pliers and scissors are examples. Students 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 users to produce forces greater than they apply.

CONNECT VOCABULARY 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.

AVOID COMMON ERRORSStudents 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.

Scoring Rubric2 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.



4.1 Angles Formed by Intersecting Lines - Lake County Angles Formed by Intersecting Lines Essential...

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