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Measuring Angles

Divider

Compass

D

Constructing an Angle Bisector

Visual Cues

Examples

Measuring Angles

Divider

Compass

D

Constructing an Angle Bisector

Visual Cues

Examples

1.
16.2 Name Class Date

Angle Measures and 16.2 Angle Measures and

Angle Bisectors Angle Bisectors

Essential Question: How is measuring an angle similar to and different from measuring a

line segment? Resource

Common Core Math Standards Locker

The student is expected to:

COMMON

CORE G-CO.A.1 Explore Constructing a Copy of an Angle

Know precise definitions of angle ... based on the undefined notions of Start with a point X and use a compass and straightedge to construct a copy of ∠S.

… distance around a circular arc. Also G-CO.D.12

Z

Mathematical Practices

COMMON

CORE MP.5 Using Tools S

X Y

Language Objective A Use a straightedge to draw a ray with endpoint X.

Work with a partner to play “angle charades.”

B Place the point of your compass on S and D Place the point of the compass on T and open it

draw an arc that intersects both sides of the to the distance TU.

angle. Label the points of intersection T and U.

U U

Essential Question: How is measuring

an angle similar to and different from S T S T

measuring a line segment?

C Without adjusting the compass, place the point E Without adjusting the compass, place the point

Possible answer: In both cases, the measure is

© Houghton Mifflin Harcourt Publishing Company

of the compass on X and draw an arc that of the compass on Y and draw an arc. Label the

undefined until a unit is chosen. Angles may be intersects the ray. Label the intersection Y. intersection with the first arc Z.

measured in degrees; there are 360° in a circle. The →

F ‾ .

Use a straightedge to draw XZ

tool for measuring an angle in degrees is a ∠X is a copy of ∠S.

protractor. Line segments are measured using linear

Reflect

units, such as centimeters or inches. The tool for → →

1. If you could place the angle you drew on top of ∠S so that XY ‾ ,

‾ coincides with ST

→

measuring a line segment is a ruler. what would be true about XZ ‾ ? Explain.

→ →

‾ would coincide with SU

XZ ‾ . Since the angles are copies of each other, the rays in each

angle form the same opening.

PREVIEW: LESSON 2. Discussion Is it possible to do the construction with a compass that is stuck open

PERFORMANCE TASK to a fixed distance? Why or why not?

No; you could use the compass to make the required arcs in Steps B and C, but you would

View the online Engage. Discuss the photo and the

not be able to adjust the opening of the compass as required in Step D.

fact that 60° and 40° stands are available, but that a Module 16 be made throu

gh “File info” 789 Lesson 2

ges must

customer wants a 50° stand. Then preview the Lesson

EDIT--Chan

DO NOT Key=NL-B;CA-B

Correction

Date

Class

an d

Measures

Name

16.2 Angle Bisectors

Performance Task. Angle

Essential

Quest ion: How

is measu

line segme

nt?

ring an angle

ions of angle

similar to

… based

and differe

on the undefi

nt from measu

ned notion

ring a

s of … distanc

e

Resource

Locker

HARDCOVER PAGES 625634

precise definit

COMMON G-CO.A.1 Know r arc. Also G-CO.D.12

an Angle

CORE

circula

around a

a Copy of

structing copy of ∠S.

IN1_MNLESE389762_U7M16L2 789 15/09/14 11:38 PM

Con tedge to constr

uct a

Explore and straigh Z

compass

and use a

Turn to these pages to

a point X

Start with

Y

X

S

int X. open it

find this lesson in the

a ray with endpo ss on T and

tedge to draw point of the compa

Use a straigh Place the

on S and

to the distan

ce TU.

compass

point of your of the

Place the cts both sides and U.

draw an arc

that interse

of interse

ction T

U

hardcover student

the points

angle. Label

U

S T

point

ss, place the the

T the compa

edition.

S adjusting Label

Without draw an arc.

the point ss on Y and

ss, place of the compa arc Z.

the compa with the first

adjusting draw an arc

that intersection

Without →

ss on X and intersection Y. ‾ .

XZ

of the compa the tedge to draw

the ray. Label

y

Use a straigh of ∠S.

g Compan

intersects ∠X is a copy

Publishin

→

→ coincides with ST ‾ ,

Harcour t

‾

XY

∠S so that rays in each

on top of other, the

Reflect you drew

place the

angle→ s of each

n Mifflin

If you could true about XZ ‾ → ? Explain. s are copie

1. the angle

what would

be

with SU ‾ . Since

© Houghto

→ would coincide

‾

XZ

same open

ing. stuck open

the ss that is

angle form with a compa would

construction but you

to do the B and C,

Is it possible not? red arcs in Steps

ssion Why or why the requi

2. Discu distance? ass to make in Step D. Lesson 2

to a fixed the comp as required

could use compass

No; you ing of the

the open

to adjust 789

not be able

11:38 PM

15/09/14

Module 16

6L2 789

62_U7M1

ESE3897

IN1_MNL

789 Lesson 16.2

Angle Measures and 16.2 Angle Measures and

Angle Bisectors Angle Bisectors

Essential Question: How is measuring an angle similar to and different from measuring a

line segment? Resource

Common Core Math Standards Locker

The student is expected to:

COMMON

CORE G-CO.A.1 Explore Constructing a Copy of an Angle

Know precise definitions of angle ... based on the undefined notions of Start with a point X and use a compass and straightedge to construct a copy of ∠S.

… distance around a circular arc. Also G-CO.D.12

Z

Mathematical Practices

COMMON

CORE MP.5 Using Tools S

X Y

Language Objective A Use a straightedge to draw a ray with endpoint X.

Work with a partner to play “angle charades.”

B Place the point of your compass on S and D Place the point of the compass on T and open it

draw an arc that intersects both sides of the to the distance TU.

angle. Label the points of intersection T and U.

U U

Essential Question: How is measuring

an angle similar to and different from S T S T

measuring a line segment?

C Without adjusting the compass, place the point E Without adjusting the compass, place the point

Possible answer: In both cases, the measure is

© Houghton Mifflin Harcourt Publishing Company

of the compass on X and draw an arc that of the compass on Y and draw an arc. Label the

undefined until a unit is chosen. Angles may be intersects the ray. Label the intersection Y. intersection with the first arc Z.

measured in degrees; there are 360° in a circle. The →

F ‾ .

Use a straightedge to draw XZ

tool for measuring an angle in degrees is a ∠X is a copy of ∠S.

protractor. Line segments are measured using linear

Reflect

units, such as centimeters or inches. The tool for → →

1. If you could place the angle you drew on top of ∠S so that XY ‾ ,

‾ coincides with ST

→

measuring a line segment is a ruler. what would be true about XZ ‾ ? Explain.

→ →

‾ would coincide with SU

XZ ‾ . Since the angles are copies of each other, the rays in each

angle form the same opening.

PREVIEW: LESSON 2. Discussion Is it possible to do the construction with a compass that is stuck open

PERFORMANCE TASK to a fixed distance? Why or why not?

No; you could use the compass to make the required arcs in Steps B and C, but you would

View the online Engage. Discuss the photo and the

not be able to adjust the opening of the compass as required in Step D.

fact that 60° and 40° stands are available, but that a Module 16 be made throu

gh “File info” 789 Lesson 2

ges must

customer wants a 50° stand. Then preview the Lesson

EDIT--Chan

DO NOT Key=NL-B;CA-B

Correction

Date

Class

an d

Measures

Name

16.2 Angle Bisectors

Performance Task. Angle

Essential

Quest ion: How

is measu

line segme

nt?

ring an angle

ions of angle

similar to

… based

and differe

on the undefi

nt from measu

ned notion

ring a

s of … distanc

e

Resource

Locker

HARDCOVER PAGES 625634

precise definit

COMMON G-CO.A.1 Know r arc. Also G-CO.D.12

an Angle

CORE

circula

around a

a Copy of

structing copy of ∠S.

IN1_MNLESE389762_U7M16L2 789 15/09/14 11:38 PM

Con tedge to constr

uct a

Explore and straigh Z

compass

and use a

Turn to these pages to

a point X

Start with

Y

X

S

int X. open it

find this lesson in the

a ray with endpo ss on T and

tedge to draw point of the compa

Use a straigh Place the

on S and

to the distan

ce TU.

compass

point of your of the

Place the cts both sides and U.

draw an arc

that interse

of interse

ction T

U

hardcover student

the points

angle. Label

U

S T

point

ss, place the the

T the compa

edition.

S adjusting Label

Without draw an arc.

the point ss on Y and

ss, place of the compa arc Z.

the compa with the first

adjusting draw an arc

that intersection

Without →

ss on X and intersection Y. ‾ .

XZ

of the compa the tedge to draw

the ray. Label

y

Use a straigh of ∠S.

g Compan

intersects ∠X is a copy

Publishin

→

→ coincides with ST ‾ ,

Harcour t

‾

XY

∠S so that rays in each

on top of other, the

Reflect you drew

place the

angle→ s of each

n Mifflin

If you could true about XZ ‾ → ? Explain. s are copie

1. the angle

what would

be

with SU ‾ . Since

© Houghto

→ would coincide

‾

XZ

same open

ing. stuck open

the ss that is

angle form with a compa would

construction but you

to do the B and C,

Is it possible not? red arcs in Steps

ssion Why or why the requi

2. Discu distance? ass to make in Step D. Lesson 2

to a fixed the comp as required

could use compass

No; you ing of the

the open

to adjust 789

not be able

11:38 PM

15/09/14

Module 16

6L2 789

62_U7M1

ESE3897

IN1_MNL

789 Lesson 16.2

2.
Explain 1 Naming Angles and Parts of an Angle

An angle is a figure formed by two rays with the same endpoint.

The common endpoint is the vertex of the angle.

EXPLORE

The rays are the sides of the angle.

Constructing a Copy of an Angle

Example 1 Draw or name the given angle.

∠PQR P

QUESTIONING STRATEGIES

When an angle is named with three letters, the middle letter is the

vertex. So, the vertex of angle ∠PQR is point Q. Do the rays of the angle you construct need to

Q

The sides of the angle are two rays with common endpoint Q. So, R be the same length as the rays of the given

→ →

‾ and QR

the sides of the angle are QP ‾ .

angle? Why or why not? No; the measure of the

Draw and label the angle as shown. angle is determined only by the size of the opening

between the rays, not by the lengths of the rays.

J

When you draw the initial arc that intersects

1

the side of the angle to be copied, does it

L K

matter how wide you open the compass?

Explain. No, as long as the arc intersects both sides

The vertex of the angle shown is point K . A name for the angle is ∠ K .

of the angle, it doesn’t matter.

The vertex must be in the middle, so two more names for the angle are ∠ J K L

and ∠ L K J .

The angle is numbered, so another name is ∠ 1 . INTEGRATE MATHEMATICAL

PRACTICES

Reflect

Focus on Modeling

3. Without seeing a figure, is it possible to give another name for ∠MKG?

If so, what is it? If not, why not? MP.4 Have students practice constructing both

© Houghton Mifflin Harcourt Publishing Company

Yes; ∠GKM acute and obtuse angles.

Your Turn

Use the figure for 4–5. B C

4. Name ∠2 in as many different ways as possible.

2

3

4

EXPLAIN 1

∠AEB, ∠BEA A E D

Naming Angles and Parts of an Angle

5. Use a compass and straightedge to copy ∠BEC.

CONNECT VOCABULARY

Connect the word degree to the idea of measurement.

A degree in science may be a measure of temperature

in units known as Fahrenheit or Celsius. Degree in

Module 16 790 Lesson 2

this context is the measure of an angle. Ask how

PROFESSIONAL DEVELOPMENT many degrees are in a straight angle, a right angle,

IN1_MNLESE389762_U7M16L2 790 4/19/14 10:34 AM

and so on.

Math Background

Compass and straightedge constructions date to ancient Greece. In fact, one of the

classic problems of ancient Greek mathematics was the trisection of an angle. That

is, using a compass and straightedge, is it possible to construct an angle whose

measure is one-third that of an arbitrary given angle? It was not until 1837 that

this construction was proven to be impossible. On the other hand, it is a

straightforward task to bisect any angle, and students learn this fundamental

construction in this lesson.

Angle Measures and Angle Bisectors 790

An angle is a figure formed by two rays with the same endpoint.

The common endpoint is the vertex of the angle.

EXPLORE

The rays are the sides of the angle.

Constructing a Copy of an Angle

Example 1 Draw or name the given angle.

∠PQR P

QUESTIONING STRATEGIES

When an angle is named with three letters, the middle letter is the

vertex. So, the vertex of angle ∠PQR is point Q. Do the rays of the angle you construct need to

Q

The sides of the angle are two rays with common endpoint Q. So, R be the same length as the rays of the given

→ →

‾ and QR

the sides of the angle are QP ‾ .

angle? Why or why not? No; the measure of the

Draw and label the angle as shown. angle is determined only by the size of the opening

between the rays, not by the lengths of the rays.

J

When you draw the initial arc that intersects

1

the side of the angle to be copied, does it

L K

matter how wide you open the compass?

Explain. No, as long as the arc intersects both sides

The vertex of the angle shown is point K . A name for the angle is ∠ K .

of the angle, it doesn’t matter.

The vertex must be in the middle, so two more names for the angle are ∠ J K L

and ∠ L K J .

The angle is numbered, so another name is ∠ 1 . INTEGRATE MATHEMATICAL

PRACTICES

Reflect

Focus on Modeling

3. Without seeing a figure, is it possible to give another name for ∠MKG?

If so, what is it? If not, why not? MP.4 Have students practice constructing both

© Houghton Mifflin Harcourt Publishing Company

Yes; ∠GKM acute and obtuse angles.

Your Turn

Use the figure for 4–5. B C

4. Name ∠2 in as many different ways as possible.

2

3

4

EXPLAIN 1

∠AEB, ∠BEA A E D

Naming Angles and Parts of an Angle

5. Use a compass and straightedge to copy ∠BEC.

CONNECT VOCABULARY

Connect the word degree to the idea of measurement.

A degree in science may be a measure of temperature

in units known as Fahrenheit or Celsius. Degree in

Module 16 790 Lesson 2

this context is the measure of an angle. Ask how

PROFESSIONAL DEVELOPMENT many degrees are in a straight angle, a right angle,

IN1_MNLESE389762_U7M16L2 790 4/19/14 10:34 AM

and so on.

Math Background

Compass and straightedge constructions date to ancient Greece. In fact, one of the

classic problems of ancient Greek mathematics was the trisection of an angle. That

is, using a compass and straightedge, is it possible to construct an angle whose

measure is one-third that of an arbitrary given angle? It was not until 1837 that

this construction was proven to be impossible. On the other hand, it is a

straightforward task to bisect any angle, and students learn this fundamental

construction in this lesson.

Angle Measures and Angle Bisectors 790

3.
Explain 2 Measuring Angles

QUESTIONING STRATEGIES The distance around a circular arc is undefined until a measurement unit is chosen. Degrees (°) are a common

1

measurement unit for circular arcs. There are 360° in a circle, so an angle that measures 1° is ___ of a circle.

When an angle is named using three letters, The measure of an angle is written m∠A or m∠PQR.

360

how can you identify the vertex of the

You can classify angles by their measures.

angle? The vertex is the center letter of the

angle name. Classifying Angles

Acute Angle Right Angle Obtuse Angle Straight Angle

An angle diagram may use letters or numbers

to identify the angle. How are the diagrams

different? Letters label individual points on the

angle, while a number is inside the angle and names A

A

A

A

the entire angle. 0° < m∠A < 90° m∠A = 90° 90° < m∠A < 180° m∠A = 180°

Example 2 Use a protractor to draw an angle with the given measure.

EXPLAIN 2 53°

→

‾ .

Step 1 Use a straightedge to draw a ray, XY

Measuring Angles

X Y

AVOID COMMON ERRORS Step 2 Place your protractor on point X as shown. Locate the point along the edge of the protractor that

corresponds to 53°. Make a mark at this location and label it point Z.

Remind students to place the center mark of the

protractor on the vertex and to align one side of the Z

angle with the 0° mark. They may have to rotate

the angle or the protractor for ease of alignment.

© Houghton Mifflin Harcourt Publishing Company

On some protractors, the zero line is on the bottom X

edge, while on others, it is placed higher. Y

→

‾ . m∠ZXY = 53°.

Step 3 Draw XZ

INTEGRATE MATHEMATICAL

Z

Focus on Modeling

MP.4 Suggest that students use a straightedge, such

as an index card, to extend the rays of an angle before

they use a protractor to measure the angle. If the X Y

angle is smaller than the distance from the center

mark to the edge of the protractor, this will make it

Module 16 791 Lesson 2

easier to accurately measure the angle. Encourage

students to estimate an angle measure before COLLABORATIVE LEARNING

measuring to make sure the measurement is IN1_MNLESE389762_U7M16L2 791 4/19/14 10:34 AM

reasonable. Small Group Activity

Use pictures from magazines to find angles of different sizes. Ask students to

identify the type of angle and estimate the measure. Then have students measure

the angles with a protractor. If protractors are not available, they can use index

cards or origami paper. The edges are already at a 90° angle, and anything greater

would be an obtuse angle. A half-fold forms a 45° angle, a tri-fold approximately

30°, and so on. The pictures can be posted by classification and used for reference.

791 Lesson 16.2

QUESTIONING STRATEGIES The distance around a circular arc is undefined until a measurement unit is chosen. Degrees (°) are a common

1

measurement unit for circular arcs. There are 360° in a circle, so an angle that measures 1° is ___ of a circle.

When an angle is named using three letters, The measure of an angle is written m∠A or m∠PQR.

360

how can you identify the vertex of the

You can classify angles by their measures.

angle? The vertex is the center letter of the

angle name. Classifying Angles

Acute Angle Right Angle Obtuse Angle Straight Angle

An angle diagram may use letters or numbers

to identify the angle. How are the diagrams

different? Letters label individual points on the

angle, while a number is inside the angle and names A

A

A

A

the entire angle. 0° < m∠A < 90° m∠A = 90° 90° < m∠A < 180° m∠A = 180°

Example 2 Use a protractor to draw an angle with the given measure.

EXPLAIN 2 53°

→

‾ .

Step 1 Use a straightedge to draw a ray, XY

Measuring Angles

X Y

AVOID COMMON ERRORS Step 2 Place your protractor on point X as shown. Locate the point along the edge of the protractor that

corresponds to 53°. Make a mark at this location and label it point Z.

Remind students to place the center mark of the

protractor on the vertex and to align one side of the Z

angle with the 0° mark. They may have to rotate

the angle or the protractor for ease of alignment.

© Houghton Mifflin Harcourt Publishing Company

On some protractors, the zero line is on the bottom X

edge, while on others, it is placed higher. Y

→

‾ . m∠ZXY = 53°.

Step 3 Draw XZ

INTEGRATE MATHEMATICAL

Z

Focus on Modeling

MP.4 Suggest that students use a straightedge, such

as an index card, to extend the rays of an angle before

they use a protractor to measure the angle. If the X Y

angle is smaller than the distance from the center

mark to the edge of the protractor, this will make it

Module 16 791 Lesson 2

easier to accurately measure the angle. Encourage

students to estimate an angle measure before COLLABORATIVE LEARNING

measuring to make sure the measurement is IN1_MNLESE389762_U7M16L2 791 4/19/14 10:34 AM

reasonable. Small Group Activity

Use pictures from magazines to find angles of different sizes. Ask students to

identify the type of angle and estimate the measure. Then have students measure

the angles with a protractor. If protractors are not available, they can use index

cards or origami paper. The edges are already at a 90° angle, and anything greater

would be an obtuse angle. A half-fold forms a 45° angle, a tri-fold approximately

30°, and so on. The pictures can be posted by classification and used for reference.

791 Lesson 16.2

4.
B 138°

QUESTIONING STRATEGIES

C

If the vertex of an angle is placed on the center

A B

point of a protractor and both rays of the

→ angle lie within the measures of the protractor, does

‾ .

Step 1 Use a straightedge to draw a ray, AB

→ one of the rays have to align with the 0° mark to find

‾ is at zero.

Step 2 Place your protractor on point A so that AB

the measure of the angle? Explain. No, you can find

Step 3 Locate the point along the edge of the protractor that corresponds to 138°.

Make a mark at this location and label it point C. the absolute value of the difference of the measures

Step 4 Draw AC

→

‾ . m∠CAB = 138°.

each ray intersects to find the measure of the angle.

For example, if one ray aligns with 25° and the other

Reflect with 67°, the angle measures 42°.

6. Explain how you can use a protractor to check that the angle you constructed

in the Explore is a copy of the given angle.

Measure the given angle and the constructed angle. They should have the same measure.

EXPLAIN 3

Your Turn

Each angle can be found in the rigid frame of the bicycle. Constructing an Angle Bisector

Use a protractor to find each measure.

7. 8. J

M

CONNECT VOCABULARY

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gena73/

K The postulates for angles are similar to the postulates

N P

L for segments. The Protractor Postulate is similar to

the Ruler Postulate. It says that the measure of an

40° 105° angle is the absolute value of the difference between

the numbers matched on a protractor with the rays

Explain 3 Constructing an Angle Bisector that form the sides of the angle.

An angle bisector is a ray that divides an angle into two angles that both have the

→ A

‾ bisects ∠ABC, so m∠ABD = m∠CBD. The arcs

same measure. In the figure, BD

in the figure show equal angle measures.

D

Postulate 2: Angle Addition Postulate

Shutterstock

C

If S is in the interior of ∠PQR, then B

m∠PQR = m∠PQS + m∠SQR. S

P

R

Q

Module 16 792 Lesson 2

DIFFERENTIATE INSTRUCTION

IN1_MNLESE389762_U7M16L2 792 4/19/14 10:34 AM

Manipulatives

Have students investigate how to find the bisector of an angle using a geometric

reflecting tool. Have students draw an angle on a piece of paper. To use the tool,

place it on the vertex of the angle so that one side is reflected onto the other side.

Then draw the tool’s line. Discuss how using the reflective device is similar to

using paper folding to find the angle bisector.

Angle Measures and Angle Bisectors 792

QUESTIONING STRATEGIES

C

If the vertex of an angle is placed on the center

A B

point of a protractor and both rays of the

→ angle lie within the measures of the protractor, does

‾ .

Step 1 Use a straightedge to draw a ray, AB

→ one of the rays have to align with the 0° mark to find

‾ is at zero.

Step 2 Place your protractor on point A so that AB

the measure of the angle? Explain. No, you can find

Step 3 Locate the point along the edge of the protractor that corresponds to 138°.

Make a mark at this location and label it point C. the absolute value of the difference of the measures

Step 4 Draw AC

→

‾ . m∠CAB = 138°.

each ray intersects to find the measure of the angle.

For example, if one ray aligns with 25° and the other

Reflect with 67°, the angle measures 42°.

6. Explain how you can use a protractor to check that the angle you constructed

in the Explore is a copy of the given angle.

Measure the given angle and the constructed angle. They should have the same measure.

EXPLAIN 3

Your Turn

Each angle can be found in the rigid frame of the bicycle. Constructing an Angle Bisector

Use a protractor to find each measure.

7. 8. J

M

CONNECT VOCABULARY

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gena73/

K The postulates for angles are similar to the postulates

N P

L for segments. The Protractor Postulate is similar to

the Ruler Postulate. It says that the measure of an

40° 105° angle is the absolute value of the difference between

the numbers matched on a protractor with the rays

Explain 3 Constructing an Angle Bisector that form the sides of the angle.

An angle bisector is a ray that divides an angle into two angles that both have the

→ A

‾ bisects ∠ABC, so m∠ABD = m∠CBD. The arcs

same measure. In the figure, BD

in the figure show equal angle measures.

D

Postulate 2: Angle Addition Postulate

Shutterstock

C

If S is in the interior of ∠PQR, then B

m∠PQR = m∠PQS + m∠SQR. S

P

R

Q

Module 16 792 Lesson 2

DIFFERENTIATE INSTRUCTION

IN1_MNLESE389762_U7M16L2 792 4/19/14 10:34 AM

Manipulatives

Have students investigate how to find the bisector of an angle using a geometric

reflecting tool. Have students draw an angle on a piece of paper. To use the tool,

place it on the vertex of the angle so that one side is reflected onto the other side.

Then draw the tool’s line. Discuss how using the reflective device is similar to

using paper folding to find the angle bisector.

Angle Measures and Angle Bisectors 792

5.
Example 3 Use a compass and straightedge to construct the bisector of the given angle. Check that

the measure of each of the new angles is one-half the measure of the given angle.

AVOID COMMON ERRORS

Remind students not to change the compass setting

when they draw the intersecting arcs from each side

ray of an angle to create the angle bisector. In order

M

to help students see why this is important, you many

want to have them do a construction in which they Step 1 Place the point of your compass on point M. Step 2 Place the point of the compass on P and

Draw an arc that intersects both sides of the draw an arc in the interior of the angle.

change the compass setting between arcs. Students angle. Label the points of intersection P and Q.

will see that the resulting ray does not bisect

the angle. P P

M M

QUESTIONING STRATEGIES Q Q

If a ray divides an angle into two angles with Step 3 Without adjusting the compass, place the

→

‾ .

Step 4 Use a straightedge to draw MR

equal measures, what must be true about the point of the compass on Q and draw an arc

that intersects the last arc you drew. Label the

ray? Explain. The ray is the angle bisector of the intersection of the arcs R.

angle by the definition of an angle bisector.

P P

R R

VISUAL CUES M

M Q Q

Some students may have difficulty visualizing two

angles that have the same measure, especially if the

Step 5 Measure with a protractor to confirm that m∠PMR = m∠QMR = _12m∠PMQ.

sides of the angles are shown with rays of different 1 (54°)✓

27° = 27° = _

2

© Houghton Mifflin Harcourt Publishing Company

lengths. You may want to have students construct A B

angle copies on tracing paper. Then they can place

the copy on top of the original angle to check that

D

the measures are the same.

C

Step 1 Draw an arc centered at A that intersects both sides of the angle.

Label the points of intersection B and C.

Step 2 Draw an arc centered at B in the interior of the angle.

Step 3 Without adjusting the compass, draw an arc centered at C that intersects

the last arc you drew. Label the intersection of the arcs D.

→

Step 4 Draw AD ‾ .

_1

Step 5 Check that m∠BAD = m∠CAD = _12m∠BAC. Yes; 45° = 45° = 2 (90°)

Module 16 793 Lesson 2

LANGUAGE SUPPORT

IN1_MNLESE389762_U7M16L2 793 15/09/14 11:38 PM

Connect Vocabulary

Remind students that the prefix bi- means “two” and that the root sect means “to

cut.” They can use these cues to help them remember that an angle bisector

divides the angle into two equal parts.

793 Lesson 16.2

the measure of each of the new angles is one-half the measure of the given angle.

AVOID COMMON ERRORS

Remind students not to change the compass setting

when they draw the intersecting arcs from each side

ray of an angle to create the angle bisector. In order

M

to help students see why this is important, you many

want to have them do a construction in which they Step 1 Place the point of your compass on point M. Step 2 Place the point of the compass on P and

Draw an arc that intersects both sides of the draw an arc in the interior of the angle.

change the compass setting between arcs. Students angle. Label the points of intersection P and Q.

will see that the resulting ray does not bisect

the angle. P P

M M

QUESTIONING STRATEGIES Q Q

If a ray divides an angle into two angles with Step 3 Without adjusting the compass, place the

→

‾ .

Step 4 Use a straightedge to draw MR

equal measures, what must be true about the point of the compass on Q and draw an arc

that intersects the last arc you drew. Label the

ray? Explain. The ray is the angle bisector of the intersection of the arcs R.

angle by the definition of an angle bisector.

P P

R R

VISUAL CUES M

M Q Q

Some students may have difficulty visualizing two

angles that have the same measure, especially if the

Step 5 Measure with a protractor to confirm that m∠PMR = m∠QMR = _12m∠PMQ.

sides of the angles are shown with rays of different 1 (54°)✓

27° = 27° = _

2

© Houghton Mifflin Harcourt Publishing Company

lengths. You may want to have students construct A B

angle copies on tracing paper. Then they can place

the copy on top of the original angle to check that

D

the measures are the same.

C

Step 1 Draw an arc centered at A that intersects both sides of the angle.

Label the points of intersection B and C.

Step 2 Draw an arc centered at B in the interior of the angle.

Step 3 Without adjusting the compass, draw an arc centered at C that intersects

the last arc you drew. Label the intersection of the arcs D.

→

Step 4 Draw AD ‾ .

_1

Step 5 Check that m∠BAD = m∠CAD = _12m∠BAC. Yes; 45° = 45° = 2 (90°)

Module 16 793 Lesson 2

LANGUAGE SUPPORT

IN1_MNLESE389762_U7M16L2 793 15/09/14 11:38 PM

Connect Vocabulary

Remind students that the prefix bi- means “two” and that the root sect means “to

cut.” They can use these cues to help them remember that an angle bisector

divides the angle into two equal parts.

793 Lesson 16.2

6.
Reflect

9. Discussion Explain how you could use paper folding to construct the bisector of an angle. ELABORATE

Fold the paper so that one side of the angle lies on top of the other. Unfold the paper.

The crease is the angle bisector. INTEGRATE MATHEMATICAL

PRACTICES

Your Turn

Focus on Math Connections

Use a compass and straightedge to construct the bisector of the given angle. Check that

the measure of each of the new angles is one-half the measure of the given angle. MP.1 Remind students to record angle measures

10. 11. using a protractor in degrees by using the degree

symbol. Point out that not all angle measures are

recorded in degrees. Radians are real number units of

angle rotation. For example, π radians = 180°.

INTEGRATE TECHNOLOGY

Elaborate

Point out that a graphing calculator may need

12. What is the relationship between a segment bisector and an angle bisector?

A segment bisector divides a line segment into two segments that have the same length;

to be set to record angle measure in degrees,

since either degree or radian measure can be selected.

an angle bisector divides an angle into two angles that have the same measure.

This feature is generally used for trigonometry

calculations, however.

13. When you copy an angle, do the lengths of the segments you draw to represent the two rays affect whether

the angles have the same measure? Explain.

No; the measure of an angle depends only on the portion of a circle that the angle

encompasses, not upon the apparent length of its sides.

QUESTIONING STRATEGIES

What methods can you use to bisect an angle?

© Houghton Mifflin Harcourt Publishing Company Which method do you think is the most

14. Essential Question Check-In Many protractors have two sets of degree measures around the edge.

When you measure an angle, how do you know which of the two measures to use? accurate? Explain. You can use a compass and

Answers may vary. Sample: First determine if the angle is acute or obtuse. If the angle straightedge, paper folding, or measurement with a

is acute, use the measure between 0° and 90°. If the angle is obtuse, use the measure protractor. Possible answer: You are more likely to

between 90° and 180°. draw the bisector accurately from the vertex by

using a compass and straightedge because the

method is exact.

SUMMARIZE THE LESSON

What is the Angle Addition Postulate and how

does it relate to the bisector of an angle? If a

Module 16 794 Lesson 2

ray from the vertex of an angle divides the angle

into two parts, the sum of the measures of the parts

is equal to the measure of the whole original angle.

IN1_MNLESE389762_U7M16L2 794 4/19/14 10:34 AM

An angle bisector is a ray that divides an angle into

two equal parts.

Angle Measures and Angle Bisectors 794

9. Discussion Explain how you could use paper folding to construct the bisector of an angle. ELABORATE

Fold the paper so that one side of the angle lies on top of the other. Unfold the paper.

The crease is the angle bisector. INTEGRATE MATHEMATICAL

PRACTICES

Your Turn

Focus on Math Connections

Use a compass and straightedge to construct the bisector of the given angle. Check that

the measure of each of the new angles is one-half the measure of the given angle. MP.1 Remind students to record angle measures

10. 11. using a protractor in degrees by using the degree

symbol. Point out that not all angle measures are

recorded in degrees. Radians are real number units of

angle rotation. For example, π radians = 180°.

INTEGRATE TECHNOLOGY

Elaborate

Point out that a graphing calculator may need

12. What is the relationship between a segment bisector and an angle bisector?

A segment bisector divides a line segment into two segments that have the same length;

to be set to record angle measure in degrees,

since either degree or radian measure can be selected.

an angle bisector divides an angle into two angles that have the same measure.

This feature is generally used for trigonometry

calculations, however.

13. When you copy an angle, do the lengths of the segments you draw to represent the two rays affect whether

the angles have the same measure? Explain.

No; the measure of an angle depends only on the portion of a circle that the angle

encompasses, not upon the apparent length of its sides.

QUESTIONING STRATEGIES

What methods can you use to bisect an angle?

© Houghton Mifflin Harcourt Publishing Company Which method do you think is the most

14. Essential Question Check-In Many protractors have two sets of degree measures around the edge.

When you measure an angle, how do you know which of the two measures to use? accurate? Explain. You can use a compass and

Answers may vary. Sample: First determine if the angle is acute or obtuse. If the angle straightedge, paper folding, or measurement with a

is acute, use the measure between 0° and 90°. If the angle is obtuse, use the measure protractor. Possible answer: You are more likely to

between 90° and 180°. draw the bisector accurately from the vertex by

using a compass and straightedge because the

method is exact.

SUMMARIZE THE LESSON

What is the Angle Addition Postulate and how

does it relate to the bisector of an angle? If a

Module 16 794 Lesson 2

ray from the vertex of an angle divides the angle

into two parts, the sum of the measures of the parts

is equal to the measure of the whole original angle.

IN1_MNLESE389762_U7M16L2 794 4/19/14 10:34 AM

An angle bisector is a ray that divides an angle into

two equal parts.

Angle Measures and Angle Bisectors 794