This pdf includes the following topics:-
Constructing an Angle Bisector
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:
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
CORE MP.5 Using Tools S
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.
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
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
customer wants a 50° stand. Then preview the Lesson
DO NOT Key=NL-B;CA-B
16.2 Angle Bisectors
Performance Task. Angle
Quest ion: How
ring an angle
ions of angle
on the undefi
nt from measu
s of … distanc
HARDCOVER PAGES 625634
COMMON G-CO.A.1 Know r arc. Also G-CO.D.12
a Copy of
structing copy of ∠S.
IN1_MNLESE389762_U7M16L2 789 15/09/14 11:38 PM
Con tedge to constr
Explore and straigh Z
and use a
Turn to these pages to
a point X
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
point of your of the
Place the cts both sides and U.
draw an arc
ss, place the the
T the compa
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
ss on X and intersection Y. ‾ .
of the compa the tedge to draw
the ray. Label
Use a straigh of ∠S.
intersects ∠X is a copy
→ coincides with ST ‾ ,
∠S so that rays in each
on top of other, the
Reflect you drew
angle→ s of each
If you could true about XZ ‾ → ? Explain. s are copie
1. the angle
with SU ‾ . Since
→ would coincide
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
to adjust 789
not be able
789 Lesson 16.2
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.
The rays are the sides of the angle.
Constructing a Copy of an Angle
Example 1 Draw or name the given angle.
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
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.
When you draw the initial arc that intersects
the side of the angle to be copied, does it
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
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.
Use the figure for 4–5. B C
4. Name ∠2 in as many different ways as possible.
∠AEB, ∠BEA A E D
Naming Angles and Parts of an Angle
5. Use a compass and straightedge to copy ∠BEC.
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.
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
Explain 2 Measuring Angles
QUESTIONING STRATEGIES The distance around a circular arc is undefined until a measurement unit is chosen. Degrees (°) are a common
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.
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
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
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
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
If the vertex of an angle is placed on the center
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.
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
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gena73/
K The postulates for angles are similar to the postulates
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
‾ bisects ∠ABC, so m∠ABD = m∠CBD. The arcs
same measure. In the figure, BD
in the figure show equal angle measures.
Postulate 2: Angle Addition Postulate
If S is in the interior of ∠PQR, then B
m∠PQR = m∠PQS + m∠SQR. S
Module 16 792 Lesson 2
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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
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
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
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.
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° = _
© 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
the measures are the same.
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 ‾ .
Step 5 Check that m∠BAD = m∠CAD = _12m∠BAC. Yes; 45° = 45° = 2 (90°)
Module 16 793 Lesson 2
IN1_MNLESE389762_U7M16L2 793 15/09/14 11:38 PM
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
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
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°.
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
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.
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