measuring and constructing angles · 1 e x a m p l e measuring and constructing angles going deeper...

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Copying an Angle

Construct an angle with the same measure as ∠S.

A In the space below, use a straightedge to draw a ray with endpoint X.

B Place the point of your compass on S and draw an arc that intersects both sides of the angle. Label the points of intersection T and U.

C Without adjusting the compass, place the point of the compass on X and draw an arc that intersects the ray. Label the intersection Y.

D Place the point of the compass on U and open it to the distance TU.

E Without adjusting the compass, place the point of the compass on Y and draw an arc. Label the intersection with the first arc Z.

F Use a straightedge to draw ___

› XZ.

E X A M P L E1

Measuring and Constructing AnglesGoing DeeperEssential question: Whattoolsandmethodscanyouusetocopyanangleandbisectanangle?

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.

Angles may be measured in degrees (°). There are 360° in a circle, so an angle that measures 1° is 1 ___ 360 of a circle. You write m∠A for the measure of ∠A.

Angles may be classified by their measures.

Acute Angle Right Angle Obtuse Angle Straight Angle

0° < m∠A < 90° m∠A = 90° 90° < m∠A < 180° m∠A = 180°

G-CO.4.12

A A A A

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Chapter 1 15 Lesson 3

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REFLECT

1a. How can you use a protractor to check your construction?

1b. If you draw ∠X so that its sides appear to be longer than the sides shown for ∠S, can the two angles have the same measure? Explain.

An angle bisector is a ray that divides an angle into two angles that both have the same measure. In the figure,

___ › BD bisects ∠ABC, so m∠ABD=m∠DBC.

The arcs in the figure show equal angle measures.

The following example shows how you can use a compass and straightedge to bisect an angle.

ConstructingtheBisectorofanAngle

Construct the bisector of ∠M. Work directly on the angle at right.

A Place the point of your compass on point M. Draw an arc that intersects both sides of the angle. Label the points of intersection P and Q.

B Place the point of the compass on P and draw an arc in the interior of the angle.

C Without adjusting the compass, place the point of the compass on Q and draw an arc that intersects the arc from Step B. Label the intersection of the arcs R.

D Use a straightedge to draw ____

› MR.

REFLECT

2a. Explain how you could use paper folding to construct the bisector of an angle.

E X A M P L E2G-CO.4.12

A

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BC

M

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R


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p r a c t i c e

Construct an angle with the same measure as the given angle.

1. 2. 3.

Construct the bisector of the angle.

4. 5. 6.

7. Explainhowyoucanuseacompassandstraightedgetoconstructananglethathastwicethemeasureof∠A.Thendotheconstructioninthespaceprovided.

A


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8. Explainhowyoucanuseacompassandstraightedgetoconstructan anglethathas1__4themeasureof∠B.Thendotheconstructioninthespaceprovided.

B


15

Name ________________________________________ Date __________________ Class __________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Practice Measuring and Constructing Angles

For Exercises 1 and 2, use the figure shown. 1. Use a compass and straightedge to

construct angle bisector →

DG—–

.Given that m∠EDF = 90°, find m∠EDG.

m∠EDG = ___________________

2. Use your construction of ∠EDG from Exercise 1. Construct ∠XYZ with the same measure as ∠EDG.

3. Use a straightedge to draw an acute angle. Use a compass and straightedge to copy the angle. Then bisect the copy of the angle.

4. Use a straightedge to draw an obtuse angle. Use a compass and straightedge to copy the angle. Then bisect the copy of the angle.

Name ________________________________________ Date __________________ Class __________________

<FILENAME = 010_CS10_G_MEPS710006_C01.docx>

Problem Solving Measuring and Constructing Segments

For Exercises 1–3, use the circle shown.

1. Copy the circle.

2. Explain how you can use your construction from Exercise 1 to construct two half circles with radius ST. Then construct the two half circles.

__________________________________________

__________________________________________

__________________________________________

3. Construct a segment MN that is the same length as ST . Then construct a triangle that has exactly two sides with length MN. How does the length of the third side compare with 2 • MN?

Choose the best answer.

4. Julia drew PQ on a piece of paper. She folded the paper so that point P was on top of point Q, forming a crease through PQ . She labeled the intersection of this crease and PQ point S. If PQ = 2.4 centimeters, then what is the length of QS ?

A 0.6 cm C 2.4 cm B 1.2 cm D 4.8 cm

5. Points J, K, and L lie on the same line, and point K is between J and L. Todd constructs SV with the same length as JL . Then he draws point T on SV so that ST is the same length as JK . Which statement is not true? F JK = ST G ST = SV − TV H SV = JK + KL J TV = ST + KL

ST

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1-3Name Class Date

Additional Practice


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Name ________________________________________ Date __________________ Class __________________



Practice Using Inductive Reasoning to Make Conjectures

In each figure, all possible diagonals are drawn from a single vertex. Use the figures in Exercises 1 and 2.

A B C D

1. Fill in the table.

Figure A B C D

Number of sides

Number of triangles formed

2. Use inductive reasoning to make a conjecture about the number of triangles formed when all possible diagonals are drawn from one vertex of a polygon with n sides.

_________________________________________________________________________________________

Complete each conjecture. 3. The square of any negative number is _______________.

4. The number of segments determined by n points is _______________.

Show that each conjecture is false by finding a counterexample. 5. For any integer n, n3 > 0.

_________________________________________________________________________________________

6. Each angle in a right triangle has a different measure.

_________________________________________________________________________________________

7. For many years in the United States, each bank printed its own currency. The variety of different bills led to widespread counterfeiting. By the time of the Civil War, a significant fraction of the currency in circulation was counterfeit. If one Civil War soldier had 48 bills, 16 of which were counterfeit, and another soldier had 39 bills, 13 of which were counterfeit, make a conjecture about what fraction of bills were counterfeit at the time of the Civil War.

_________________________________________________________________________________________

Name ________________________________________ Date __________________ Class __________________



A

C

B

Problem Solving Measuring and Constructing Angles

For Exercises 1–2, use the figure shown.

1. Construct the bisector of ∠B in ABC. Construct the bisectors of

∠A and ∠C.

2. What do you notice about the bisectors?

_________________________________________

_________________________________________

_________________________________________

_________________________________________

3. Construct an angle whose measure is four times as great as m∠K, shown below.

Choose the best answer.

4. Mark drew ∠PQR with measure 168°. He constructed the angle bisector →

QX—–

.Then he bisected ∠PQX by constructing →

PY—–

. What is m∠PQY?

A 21° B 42° C 84° D 168°

5. Paula bisected ∠XYZ, forming angles ∠XYW and ∠WYZ. Given that ∠XYZ is an obtuse angle, which statement cannot be true?

F m∠WYZ is less than 90°.

G m∠WYZ = m∠XYW

H m∠WYZ is greater than 90°.

J m∠XYZ = 2 • m∠XYW

K

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Problem Solving


measuring and constructing angles · 1 e x a m p l e measuring and constructing angles going deeper...

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