1-7 Basic Constructions

1-7

1. Plan

Objectives

1 To use a compass and a straightedge to construct congruent segments and congruent angles

2 To use a compass and a straightedge to bisect segments and angles

Examples

1 Constructing Congruent Segments

2 Constructing Congruent Angles

3 Constructing the Perpendicular Bisector

4 Finding Angle Measures 5 Constructing the Angle

Bisector

Math Background

Construction methods are justified by postulates such as Euclid's Fourth Postulate, that a circle can be drawn with any center and any positive radius, and by, for example, triangle congruency theorems. Compassand-straightedge constructions provide a hands-on introduction to these postulates and theorems.

More Math Background: p. 2D

Lesson Planning and Resources

See p. 2E for a list of the resources that support this lesson.

PowerPoint

Bell Ringer Practice

Check Skills You'll Need For intervention, direct students to:

Finding Segment Lengths Lesson 1-3: Example 1 Extra Skills, Word Problems,

Proof Practice, Ch. 1

Finding Angle Measures Lesson 1-4: Examples 4 and 5 Extra Skills, Word Problems,

Proof Practice, Ch. 1

44

1-7

Basic Constructions

What You'll Learn

? To use a compass and a

straightedge to construct congruent segments and congruent angles

? To use a compass and a

sraightedge to bisect segments and angles

. . . And Why

To construct the bisector of an angle to illustrate angles of incidence and reflection, as in Exercise 18

Check Skills You'll Need

GO for Help Lessons 1-5 and 1-6

In Exercises 1?6, sketch each figure). 1?6. See back of boo*k. )

1. CD

2. GH

3. AB

4. line m

5. acute &ABC

6. XY 6 ST

7. DE = 20. Point C is the midpoint of DE. Find CE. 10 8. Use a protractor to draw a 608 angle. 8?9. See back of book. 9. Use a protractor to draw a 1208 angle.

New Vocabulary ? construction ? straightedge ? compass ? perpendicular lines ? perpendicular bisector ? angle bisector

1 Constructing Segments and Angles

In a construction you use a straightedge and a compass to draw a geometric figure. A straightedge is a ruler with no markings on it. A compass is a geometric tool used to draw circles and parts of circles called arcs.

Four basic constructions involve constructing congruent segments, congruent angles, and bisectors of segments and angles.

1 EXAMPLE Constructing Congruent Segments

Construct a segment congruent to a given segment.

Given: AB Construct: CD so that CD > AB

A

B

Step 1

Draw a ray with endpoint C.

C

Step 2 Open the compass to the length of AB.

1.

Step 3

A

B

X

Y

With the same compass setting, put the compass point on point C. Draw an arc that intersects the

ray. Label the point of intersection D.

R

S

C

D

CD > AB

Quick Check 1 Use a straightedge to draw XY. Then construct RS so that RS = 2XY.

44 Chapter 1 Tools of Geometry

Special Needs L1 For Example 3 on constructing a perpendicular bisector, ask: Why does the opening of the compass need to be greater than half the length of the segment? the arcs must intersect

learning style: verbal

Below Level L2 Demonstrate the construction steps in Examples 1, 2, 3, and 5 by using a large demonstration compass. Then ask student volunteers to demonstrate the constructions.

learning style: visual

2 EXAMPLE Constructing Congruent Angles

Construct an angle congruent to a given angle.

Given: &A

Construct: &S so that &S > &A

A

For: Construction Activity Use: Interactive Textbook, 1-7

Step 1 Draw a ray with endpoint S.

Step 2 With the compass point on point A, draw an arc that intersects the sides of &A. Label the points of intersection B and C.

Step 3 With the same compass setting, put the compass point on point S. Draw an arc and label its point of intersection with the ray as R.

Step 4 Open the compass to the length BC. Keeping the same compass setting, put the compass point on R. Draw an arc to locate point T.

S

B A

C

S

R

T

S

R

Step 5 )

Draw ST. &S > &A

Quick Check 2 Construct &F with m&F = 2m&B.

T

S

R

B

B

F

21 Constructing Bisectors

Perpendicular lines are two lines that intersect to form right angles. The symbol '

means "is* per)pen* dicu) lar t*o." I)n th* e d)iagram at

the right, AB ' CD and CD ' AB .

A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments.

A

D

C

B

Real-World Connection

Perpendicular hands signal "Time out."

As you will learn in Chapter 5, there is just one line that is the perpendicular bisector of a segment in a given plane. Here is a way to construct the perpendicular bisector.

Lesson 1-7 Basic Constructions 45

2. Teach

Guided Instruction

Teaching Tip

If students have problems using a compass and straightedge, have them work in pairs to share the construction steps.

1 EXAMPLE Error Prevention

Have students make sure that their compasses are not so loose that they slip during use and make an incorrect arc. Point out that some compasses can be tightened.

PowerPoint

Additional Examples

1 Construct TW congruent to KM.

K

M

T

W

2 Construct &Y so that &Y &G.

G

Y

Advanced Learners L4

English Language Learners ELL

Have students investigate ancient and modern

Help students understand the difference between

attempts to trisect an angle.

drawing and constructing. Emphasize that when

drawing, a ruler and protractor can be used. In

constructions, only a compass and straightedge

can be used.

learning style: verbal

learning style: verbal

45

Guided Instruction

Teaching Tip Point out that the symbol for perpendicular resembles perpendicular lines. Ask: What other symbol resembles what it represents? Sample: the symbol for parallel

3 EXAMPLE

Connection to Language Arts

The word bisector contains the prefix bi-, which means two.

PowerPoint

Additional Examples

3 Open the compass less than 12AB in step 1 of Example 3. Explain why

the construction is not possible.

Sample: When the opening is

less

than

1 2

AB,

the

arcs

drawn

in

steps 1 and 2 do not intersect.

So, the perpendicular bisector

cannot be drawn.

4 WR) bisects &AWB.

m&AWR = x and

m&BWR = 4x - 48. Find m&AWB. 32

5 Construct MX) , the bisector

of &M.

M

X

M

Resources

? Daily Notetaking Guide 1-7 L3

? Daily Notetaking Guide 1-7--

Adapted Instruction

L1

3 EXAMPLE Constructing the Perpendicular Bisector

Construct the perpendicular bisector of a segment.

Given: AB * )

*)

Construct: XY so that XY ' AB

A

B

at the midpoint M of AB.

Step 1

Put the compass point on

point A and draw a long arc

as shown. Be sure the opening

is greater than 12AB.

A

Real-World Connection

Careers Architects use

construction tools to work with their designs.

Step 2 With the same compass setting, put the compass point on point B and draw another long arc. Label the points where the two arcs intersect as X and Y.

Step 3* ) Draw XY. The point of* )

intersection of AB and XY is M, the midpoint of AB.

*) XY* ') AB at the midpoint of AB,

so XY is the perpendicular bisector of AB.

A AM

Quick Check 3 Draw ST. Construct its perpendicular bisector. S

B

X B

Y X

B Y

T

An angle bisector is a ray that divides an angle into two congruent coplanar angles. Its endpoint is at the angle vertex. Within the ray, a segment with the same endpoint is also an angle bisector. You may say that the ray or segment bisects the angle.

4 EXAMPLE Finding Angle Measures )

Algebra KN bisects &JKL so that m&JKN = 5x - 25 and m&NKL = 3x + 5. Solve for x and find m&JKN.

J

K

N

L

m&JKN = m&NKL 5x - 25 = 3x + 5

5x = 3x + 30 2x = 30

x = 15 m&JKN = 5x - 25 = 5(15) - 25 = 50

Definition of angle bisector Substitute. Add 25 to each side. Subtract 3x from each side. Divide each side by 2. Substitute 15 for x.

m&JKN = 50

Quick Check 4 Find m&NKL and m&JKL. 50; 100

Closure

46 Chapter 1 Tools of Geometry

Write steps in your own words for bisecting an angle and bisecting a segment. Use drawings that show

Quick Check

1. X

Y

the arcs in each step. Check

students' work.

R

S

46

5. a.

X P

Y

Z

Exercises

1.

AB

X

Y

2.

AB

V

AB W

5 EXAMPLE Constructing the Angle Bisector

Construct the bisector of an angle.

Given: &A )

A

Construct: AX, the bisector of &A

Step 1 Put the compass point on vertex A. Draw an arc that intersects the sides of &A. Label the points of A intersection B and C.

Step 2

Put the compass point on point C and draw an arc.

With the same compass setting, draw an arc using point B. Be sure the arcs intersect. Label the point

A

where the two arcs intersect as X.

B

C

B X

C

Quick Check

Step 3 )

Draw AX.

)

AX is the bisector of &CAB.

B X

A

C

)

5 a. Draw obtuse &XYZ. Then construct its bisector YP. See margin, p. 46.

b. Explain how you can use your protractor to check your construction.

Measure lXYP and lPYZ to see that they are O.

EXERCISES

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

Practice and Problem Solving

A Practice by Example

GO

for Help

Example 1 (page 44)

In Exercises 1?8, draw a diagram similar to the given one. Then do the

construction. Check your work with a ruler or a protractor. 1?8. See margin,

1. Construct XY congruent to AB.

pp. 46?47.

2. Construct VW so that VW = 2AB.

A

B

3. Construct DE so that DE = TR + PS. 4. Construct QJ so that QJ = TR - PS. T

RP

S

Example 2

5. Construct &D so that &D > &C.

C

(page 45)

6. Construct &F so that m&F = 2m&C.

Example 3 (page 46)

Example 4 (page 46)

7. Construct the perpendicular bisector of AB.

8. Construct the perpendicular bisector of TR.

)

x2 9. Algebra GH bisects &FGI. a. Solve for x and find m&FGH. 11; 30 b. Find m&HGI. 30 c. Find m&FGI. 60 G

F (3x (4x

3) H - 14)

I

3. TR

D

PS E

4. TR

Q J PS

5. D

Lesson 1-7 Basic Constructions 47

6.

F

7.

A

B

3. Practice

Assignment Guide

1 A B 1-6, 15, 22-24, 29-32, 34

2 A B 7-14, 16-21, 25-28, 33

C Challenge

35-36

Test Prep Mixed Review

37-40 41-50

Homework Quick Check

To check students' understanding of key skills and concepts, go over Exercises 4, 14, 18, 26, 28.

Technology Tip Have students investigate what software can model compass and straightedge constructions. Some programs use a mouse and pointer to model the action of compass, straightedge, and pencil.

Exercises 2?4 If necessary, discuss ways that students can copy the segment lengths.

GPS Guided Problem Solving

L3

Enrichment

Reteaching

Adapted Practice

PraNcamte ice

Class

Practice 1-7

Find the area of each rectangle with the given base and height.

1. base: 3 ft height: 22 in.

2. base: 60 in. height: 1.5 yd

Find the circumference of each circle in terms of .

4.

5.

16

16

L4

L2

L1

Date

L3

Perimeter, Circumference, and Area

3. base: 2 m height: 120 cm

6. 3.9

? Pearson Education, Inc. All rights reserved.

Find the perimeter and area of each rectangle with the given base and height.

7. b = 7 cm, h = 6 cm

8. b = 21 cm, h = 2 cm

10. b = 17 ft, h = 3 ft

11. b = 11 m, h = 9 m

9. b = 4 in., h = 10.5 in. 12. b = 13 m, h = 7 m

Find the perimeter and area of each figure. All angles in the figures are right angles.

13. 15

14.

7

15.

4

19

4

4

2

7

2

2

4

Find the area of each circle in terms of .

16.

17.

12.5 200

8

18. p2?

19. Find the area and perimeter of rectangle ABCD with vertices A(3, 7), B(9, 7), C(9, -1), and D(3, -1).

20. Find the perimeter of PQR with vertices P(-2, 9), Q(7, -3), and R(-2, -3).

21. The circumference of a circle is 26p. Find the diameter and the radius.

8.

T

R

47

Visual Learners Exercises 10?12 Have students

draw each angle and its bisector before beginning to solve the exercise.

Connection to Algebra Exercise 25 This exercise is

important to assign. Discuss why both Lani and Denyse are correct.

Exercise 34 Some students may think the answer choices are the same. Encourage them to read the answer choices carefully looking for how each choices differs.

13.

P

Q

R

14.

T

U

15.

V

120 W

Z

4 16. Find a segment on XY

so that you can 4

construct YZ as its # bisector.

Z

X

Y

Z

X

Y

48

)

x2 Algebra For Exercises 10?12, BX bisects &ABC. Solve for x and find m&ABC. 10. m&ABX = 5x, m&XBC = 3x + 10 5; 50 11. m&ABC = 4x - 12, m&ABX = 24 15; 48 12. m&ABX = 4x - 16, m&CBX = 2x + 6 11; 56

Example 5 (page 47)

B Apply Your Skills

13. Draw acute &PQR. Then construct its bisector. See margin.

14. Draw right &TUV. Then construct its bisector. See margin.

15. Use your protractor and draw &W with m&W = 120. Construct &Z > &W. Then construct the bisector of &Z. See margin.

Ske*tch t)he *figu)re described. Explain how to constr)uct it. Then do the construction.

16. XY ' YZ 16?17. See margin. 17. ST bisecting right &PSQ

18. Optics A beam of light and a mirror

GPS can be used to study the behavior of light. Light that strikes the mirror is

Angle of

Angle of

incidence reflection C

reflected so that the angle of reflection

A

D

and the angle of incidence are

congruent. In the diagram, BC is

perpendicular to the mirror, and

&ABC has a measure of 418.

E

B

F

a. Name the angle of reflection

and find its measure. lCBD; 41

b. Find m&ABD. 82

c. Find m&ABE and m&DBF. 49; 49

19. Use a straightedge and protractor. a?b. See back of book. a. Draw a mirror and a light beam striking the mirror and reflecting from it. b. Construct the bisector of the angle formed by the incoming and reflected light beams. Label the angles of incidence and reflection.

21b. Infinitely many; there's only 1 midpt. but there exist infinitely many lines through the midpoint. A segment has exactly one # bisecting line because there can be only one line # to a segment at its midpt.

c. There are an infinite number of lines in space that are # to a segment at its midpt. The lines are coplanar.

20. Open-Ended Snoopy can draw squares with his compass. You can only draw circles. You can, however, construct a square. Explain how to do this. Use sketches if needed. Then do the construction. See back of book.

21. Answer these questions about a segment in a plane. Explain each answer. a. How many midpoints does the segment have? See back of book. b. How many bisectors does it have? How many lines in the plane are its perpendicular bisectors? b?c. See left. c. How many lines in space are its perpendicular bisectors?

GO nline

For Exercises 22?24, copy &1 and &2. 22?24. See back of book.

Homework Help

22. Construct &B so that m&B = m&1 + m&2.

1

2

Visit: Web Code: aue-0107

23. Construct &C so that m&C = m&1 - m&2.

24. Construct &D so that m&D = 2m&2.

48 Chapter 1 Tools of Geometry

4 17. Find a segment on SQ

so that you can 4

construct SP as its

# bisector. Then bisect

/PSQ.

P

U

S

Q

P

T

S

Q

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