block 1 ~ lines and angles - oregon focusblock 1 ~ lines and angles ~ angle pairs 1 block 1 ~ lines...

Block 1 ~ Lines And Angles ~ Angle Pairs 1

BLoCK 1 ~ LInes And AngLesangLe pairs

VerticAl Angles

word wAll

Vertex

Acute Angle

oBtuse Angle

suPPlementAry Angles

trAnsVersAl

sAme-side interior Angles

deGree

corresPonding Angles

AlternAte interior Angles

AlternAte exterior Angles

right Angle

strAight Angle

Angle

Protractor

AdJAcent Angles

comPlementAry Angles

congruent

lineAr PAir

rAy

Lesson 1 MeasUring and naMing angLes -------------------------------------- 3

Lesson 2 cLassiFYing angLes -------------------------------------------------- 8

Explore! Classify an AngleLesson 3 coMPLeMenTarY and sUPPLeMenTarY angLes ------------------------- 13

Explore! Complementary vs SupplementaryLesson 4 verTicaL angLes ----------------------------------------------------- 18

Explore! The Vertical Angle RelationshipLesson 5 aLTernaTe exTerior and inTerior angLes ---------------------------- 23

Explore! Alternate Exterior and Alternate InteriorLesson 6 corresPonding and saMe-side inTerior angLes --------------------- 29

Explore! More Angle Pairsreview BLock 1 ~ angLe Pairs ----------------------------------------------- 35

2 Block 1 ~ Angle Pairs ~ Tic - Tac - Toe

BLoCK 1 ~ AngLe PAIrstic - tac - tOe

PuZZling Angles

Find angle measures in a complex diagram.

See page for details.

Bisecting Angles

Use two types of constructions to

bisect angles.


FliP BooK

Create a fl ip book which describes

special angle pairs.


croSSword

Make a crossword using vocabulary from this block.


ProtrActor guide

Write a guide for using a protractor.


trAnsVersAl collAge

Find or take pictures of transversals and

display them.


Angle Art

Create an original piece of artwork with lines and angles.


BooK oF Poetry

Write poems about special angle pairs. Make an

illustrated poetry booklet.


duPlicAting Angles

Use a compass and straightedge to

duplicate angles.


Special Angle Pairs

Lesson 1 ~ Measuring And Naming Angles 3

measuring and naming angLes

Lesson 1

Angles are used in construction, architecture, graphic design, aerospace, art, machining and manufacturing, as well as many other fi elds. An angle is formed by two rays with a common endpoint. A ray has one endpoint and extends forever in one direction.

Ray NA is written ___

› NA . Ray NG is written

___ › NG . Th e fi rst point in the name of a

ray is the endpoint. Th e vertex of an angle is the common point of both rays. N is the vertex of this angle.

When three points are used to name an angle, the vertex is written in the middle of the name. Th e vertex can be written as the name of an angle when it is the vertex for only one angle.

Th ree ways to name the angle formed by ___

› NA and

___ › NG are ∠ANG, ∠GNA and ∠N.

give 4 diff erent names for the given angle.

1. ∠PAL 2. ∠LAP 3. ∠A 4. ∠1

Is ∠W another name for ∠nWe? explain.

No, it is not clear whether ∠W refers to ∠NWE, ∠SWE or ∠NWS.

Adjacent angles are two angles that share a ray. In Example 2, ∠NWE and ∠SWE share ___

› WE .

Th is means ∠NWE and ∠SWE are adjacent angles.

exampLe 1

solutions

exampLe 2

solution

P

A

L

1

E

WN

S

A

N

G

___ ›

NG (ray NG)

___ ›

NA (ra

y NA)

4 Lesson 1 ~ Measuring And Naming Angles

Identify at least one additional name for each angle. Write using proper angle notation. a. ∠1

b. ∠TGI

c. ∠RGE

a. ∠TGE or ∠EGT

b. ∠2 or ∠IGT

c. ∠EGR or ∠3

A protractor is a tool used to measure angles. Angles are measured in units called degrees.

Th e “m” in front of an angle measure is notation for the word “measure”. Th e statement in Figure 1 below reads, “Th e measure of ∠ABC is equal to sixty degrees.”

Figure 1 Figure 2 Figure 3

m∠ABC = 60° m∠PQR = 142° m∠ZYX = 74°

exampLe 3

solutions

P

Q R142°

Z

X Y74°

A

B C60°

TI

E

G

R

3

21


use a protractor to measure each angle. a. b.

a. b.

m∠ABC = 53° m∠5 = 132°

exercises

give two different names for each angle. 1. 2. 3.

sketch a diagram to represent each angle. 4. ∠DOG 5. ∠CUB also called ∠4

6. ∠PAL also called ∠2 7. ∠1 and ∠2 are adjacent angles

8. ∠XYZ and ∠XYU are adjacent angles 9. ∠HOT is approximately 90°

use each protractor to determine the measure of the angle.

10. 11.

exampLe 4

solutions

5

5

B

C

A

B

C

A

B

AT

KJ

LQ

U

P

L

M S B I

G

6 Lesson 1 ~ Measuring And Naming Angles

use a protractor to measure each angle to the nearest degree.

12. 13. 14.

15. 16. 17.

use a protractor to draw each angle. Label the angle(s).

18. m∠SAM = 34° 19. m∠YAK = 115°

20. m∠CAT = 167° 21. an 80° angle with ___

› PQ and

___ › PR

22. two 35° angles with the same vertex 23. two adjacent angles that are 50° and 100°

use the diagram below to name an angle with the specified measure.

24. 90° 25. 50° 26. 17°

27. 145° 28. 163° 29. 130°

A

C

B

D

E

M

G

D

EF

M

AN

G

UR

3

P

IV

1


tic-tAc-toe ~ duPl icAti ng Angle s

Constructions are part of geometry. A geometric construction is made by using a compass and straightedge. Follow the steps to duplicate ∠ABC on a piece of notebook paper.

step 1: Trace ∠ABC on your paper.

step 2: Use a straightedge to draw a ray. Th is will be one side of the duplicate angle.

step 3: Place the stylus or sharp point of a compass on the vertex of the traced angle. Draw an arc on the angle.

step 4: Without changing the setting on the compass, place the stylus on the endpoint of the duplicate ray and draw an arc.

step 5: Use the compass to measure the width of the arc drawn on the original angle. Place the stylus on the intersection of a side and the arc. Adjust the compass so the pencil is touching the other intersection point.

step 6: Without changing the setting on the compass, place the stylus on the intersection of the arc and duplicate ray. Make a small arc intersecting the larger arc.

step 7: Use a straightedge to connect the endpoint of the duplicate ray. Th is is the second ray needed to complete the duplication of ∠ABC.

1. Use a protractor to draw a 60° angle.

2. Using only a compass and straightedge, duplicate the 60° angle.

3. Measure the duplicated angle with a protractor to check accuracy.

4. Repeat these steps on a 25°, 128° and a 160° angle.

5. Which step is the most diffi cult for you in this process? Why?

Place stylus here.

B

A

C

Place stylus here.

8 Lesson 2 ~ Classifying Angles

cLassiFying angLes

Lesson 2

In Lesson 1 you named and measured angles. Angles can be classifi ed into groups by their degree measure.

step 1: Use a protractor to measure the angles in each group. Record each measurement.

grouP A grouP B grouP C

step 2: Answer each question for each group. a. How are the angles in each group alike? b. What do you notice about the degree measures of the angles in each group? step 3: Write at least two sentences describing each group of angles. Include information about their degree measures.

step 4: Compare what you wrote in step 3 to the defi nitions given below. Identify which defi nition is appropriate for each group. Acute angle – an angle that measures more than 0° and less than 90° Right angle – an angle that measures exactly 90° Obtuse angle – an angle that measures more than 90° but less than 180°

expLOre! cLassiFy an angLe

Lesson 2 ~ Classifying Angles 9

Th ere are four classifi cations of angles based on their degree measure. In the Explore!, you learned about acute, obtuse and right angles. An angle that has a measure of 180° is called a straight angle.

rig

htAngle

strAight Angle

Right angles are oft en identifi ed by drawing a small square in the vertex of the angle. If a square is present in the vertex of an angle, the angle measures 90⁰.

Classify each angle by measuring it with a protractor.a. b.

c. d.

a. m∠CAT = 64° so the angle is acute. b. m∠JIL = 110° so the angle is obtuse.c. Th e box drawn at the vertex shows that m∠FOX = 90°.d. m∠DOG = 180° so it is a straight angle.

Angles with equal measures are congruent. Th e symbol for congruent is ≅. Congruent angles are identifi ed in diagrams with congruence marks. Th e two marks on the arc inside each angle are congruence marks. Th ey show ∠RED ≅ ∠BLU. Th is is read, “Angle RED is congruent to angle BLU.” ∠X is congruent to ∠Y because each measures 90°. Each angle is a right angle as indicated by the square in each vertex.

exampLe 1

solutions

A

C

T

J

I L

OF

X

B L

U

D

E

RX

Y

D O G


sketch a diagram of congruent and adjacent angles. The angles share

___ › OC which makes them adjacent.

The congruence marks indicate the angles are congruent.

use the information in the diagram to write an equation. solve for x.

Congruence marks show the angles are congruent. ∠SIM ≅ ∠PLE Write the equation. 5x − 11 = 39 Add 11 to both sides of the equation. +11 +11 Divide both sides of the equation by 5. 5x __ 5 = 50 __ 5

x = 10 ☑ Check the solution by substituting 10 for x. 5x − 11 = 39 5(10) − 11 ?= 39 50 − 11 ?= 39 39 = 39

∠JAK is congruent to ∠hIL. The measure of ∠JAK = (12 − 3x)° and the measure of ∠hIL = (44 − x)°. solve for x. Then find the degree measure of each angle. Write an equation showing the angles ∠JAK ≅ ∠HIL have equal measures. 12 − 3x = 44 − xAdd x to each side of the equation. +x +x 12 − 2x = 44 Subtract 12 from each side of the equation. −12 −12 Divide each side of the equation by −2. −2x ___ −2 = 32 __ −2 x = −16

Write the given expression for each angle. m∠JAK = 12 − 3x m∠HIL = 44 − x Substitute −16 for x. = 12 − 3(−16) = 44 − (−16) Simplify. = 12 + 48 = 44 + 16 Add. = 60 = 60

m∠JAK = 60° and m∠HIL = 60°. Both angles are equal which verifies that they are congruent.

exampLe 2

solution C

OB

exampLe 3

solution

S

IM

(5x − 11)° E

P

L

39°

exampLe 4

solution

D

Lesson 2 ~ Classifying Angles 11

exercises

estimate the degree measure for each angle. Classify each angle as acute, obtuse, right or straight.

1. 2. 3.

4. 5. 6.

sketch a diagram for each description. Label each angle.

7. two congruent angles 8. ∠GUY is obtuse

9. ∠GAL is acute 10. ∠JKL ≅ ∠POM 11. a right angle that can be identified using 3 names 12. two adjacent, right angles

13. two congruent angles that share a vertex 14. ∠PQR is straight

use each diagram to solve for x. 15. 16. m∠PQR = 124° 17.

18. 19. ∠ROY ≅ ∠MAN 20. m∠XYZ = m∠ABC m∠XYZ = (5 + 2x)° m∠ABC = (3x + 1)°

21. ∠JAM and ∠GEM are congruent angles. The measure of ∠JAM is (8x + 5)° and m∠GEM = (x + 75)°. a. Solve for x. b. Find the measure of each angle.

22. ∠SML is an acute angle. The measure of ∠SML = (x − 7)°. What must x be less than?

R

OY

(7x + 3)°M

A

N(5x + 13

)°

30°

(4x + 6)°||

||

|| 60° (x + 12)°||

(2x − 10)°(3x − 2)°

P Q

R


23. m∠LRG = (4x + 22)° a. If ∠LRG is a right angle, what must x equal?b. If ∠LRG is an acute angle, what must x be less than?c. If ∠LRG is an obtuse angle, what must x be greater than?

review

sketch a diagram to represent each fi gure. 24. ∠RMP 25. ∠PIE also called ∠3 26. ∠5 and ∠6 which are adjacent use a protractor to measure each angle.

27. 28. 29.

use a protractor to draw an angle with the given measure. 30. 67° 31. 135°

32. 180° 33. 11°

tic-tAc-toe ~ Protr Actor gu i de

step 1: Create a user’s guide describing how to measure and draw angles with a protractor. Your teacher may use the guide to refresh a substitute teacher or for students who are absent the day of the lesson.

step 2: Create a worksheet that can be completed using the guide.

step 3: Make an answer key for the worksheet.

Lesson 3 ~ Complementary And Supplementary Angles 13

cOmpLementary and suppLementary angLes

Lesson 3

Individual angles are classifi ed as acute, right, obtuse or straight. Special pairs of angles can also be classifi ed. In Lessons 3-6, special pairs of angles and their relationships will be examined.

step 1: Look at the angles in the chart. a. What similarities do you notice about the pairs of angles called supplementary angles? b. What similarities do you notice about the pairs of angles called complementary angles?

step 2: Write a defi nition for supplementary angles.

step 3: Write a defi nition for complementary angles.

step 4: Give at least two examples of angle measures for each type of angle pair listed below. a. Supplementary angles b. Complementary angles

expLOre! cOmpLementary vs suppLementary

suPPLeMentArY AngLes CoMPLeMentArY AngLes

∠CAT and ∠DOG are supplementary.

A

TC D

O G

∠LEF and ∠RGT are supplementary.

L

E F

R

GT

∠6 and ∠7 are supplementary.m∠6 = 80° and m∠7 = 100° ∠8 and ∠9 are

complementary.

∠CAR and ∠WLK are complementary.LK

W20°

A

RC

70°

∠ROM and ∠MOE are complementary.

M

E

R

O

∠1 and ∠2 are complementary.

21

∠1 and ∠2 are supplementary.2 1

m∠8 = 45° and m∠9 = 45°

40°140°

14 Lesson 3 ~ Complementary And Supplementary Angles

Complementary and supplementary angles are special pairs of angles. Complementary angles are two angles with a sum of 90°. Two angles with a sum of 180° are called supplementary angles. These special pairs of angles may or may not be adjacent.

use the diagram to find m∠PAr.

m∠PAR and m∠TAR are supplementary.

Supplementary angles have a sum of 180°. m∠PAR + 45° = 180° Subtract 45 from both sides of the equation. −45° −45° m∠PAR = 135°m∠PAR is 135°

☑ Check the solution. 135° + 45° ?= 180° 180° = 180°

∠grA and ∠Ins are supplementary. a. Write an equation to solve for x. b. determine the measure of each angle.

a. Supplementary angles have a sum of 180°. m∠GRA + m∠INS = 180° Write an equation. (2x + 4) + (3x + 1) = 180 Combine like terms. 5x + 5 = 180 Subtract 5 from each side. −5 −5 Divide each side by 5. 5x __

5 = 175 ___ 5

x = 35

b. Write the given expression for each angle. m∠GRA = (2x + 4)° m∠INS = (3x + 1)° Substitute 35 for x. = (2(35) + 4)° = (3(35) + 1)° Multiply. = (70 + 4)° = (105 + 1)° Add. = 74° = 106°

☑ m∠GRA + m∠INS = 180° 74° + 106° ?= 180° 180° = 180°

The measure of ∠GRA is 74°. The measure of ∠INS is 106°.

exampLe 1

solution

exampLe 2

solutions

G

R

A(2x + 4)°

I

N S(3x + 1)°

P A t

r

45°

use the diagram to write an equation. solve for x.

Complementary angles have a sum of 90°. m∠HOM + m∠MOE = 90° Substitute the degreee measures. 62 + (x + 5) = 90 Combine the like terms. x + 67 = 90 Subtract 67 from each side of the equation. −67 −67 x = 23°

☑ Check the solution. 62 + (23 + 5) ?= 90 62 + 28 ?= 90 90 = 90The value of x is 23.

∠1 and ∠2 are complementary angles. The measure of ∠1 = (3x + 4)° and m∠2 = (x + 6)°.a. draw a diagram.b. Write an equation and solve for x.c. Find ∠1 and ∠2.

a.

or

b. Complementary angles have a sum of 90°. m∠1 + m∠2 = 90° Substitute the degree measures. (3x + 4) + (x + 6) = 90 Combine like terms on the same side of the equation. 4x + 10 = 90 Subtract 10 from each side of the equation. −10 −10 Divide by 4 on each side of the equation. 4x __ 4 = 80 __ 4 x = 20

c. Write the given expression for each angle. m∠1 = (3x + 4)° m∠2 = (x + 6)° Substitute 20 for x. = (3(20) + 4) = (20 + 6) Multiply. = (60 + 4) Add. = 64° = 26°

☑ Check the solution. 64° + 26° ?= 90° 90 = 90 m∠1 = 64° and m∠2 = 26°.


exampLe 3

solution

exampLe 4

solutions

(3x + 4)° (x + 6)°(x + 6)°

(3x + 4)°

eo

M

h

(x + 5)°62°

16 Lesson 3 ~ Complementary And Supplementary Angles

exercises

Identify each pair of angles as complementary, supplementary or neither. 1. 2. 3.

4. 5. 6.

7. m∠1 and m∠2 sum to 181°. 8. ∠A and ∠M have a sum of 90°.

Write an equation for each description. solve for x. Check your solution. 9. ∠A and ∠B are complementary. 10.

11. 12.

13. 14.

15. ∠MAN and ∠MAP are supplementary. The measure of ∠MAN is 57°. What is the measure of ∠MAP?

16. ∠5 and ∠7 are complementary angles. Find the measure of ∠7 if m∠5 = 47°.

17. The complement of ∠Q is 31°. Find m∠Q.

18. The supplement of ∠U is 62°. What is m∠U?

x°

AB

x° 41°

34°

56°

110° 70°

58°

122°

122°

20°

52°38°

42°

x°x°

(7x − 5)

°

(x + 3)°

2x° x°

70° (4 + 2x)°


Find the measure of each angle in exercises 19-23. Check your solution.

19. ∠1 and ∠2 are supplementary; m∠1 = 3x° and m∠2 = 3x° 20. ∠W And ∠C are complementary; m∠W = (47 + 3x)° and m∠C = (10 + 8x)°

21. ∠G and ∠H are supplementary; m∠G = (x + 4)° and m∠H = (4x + 11)°

22. ∠V and ∠W are supplementary; ∠V is (12 + 3x)° and m∠W = (33 + 2x)°

23. ∠1 and ∠2 are complementary; m∠1 = 4x° and m∠2 = (x + 8)°

24. Use the fi gure on the right. a. Name an obtuse angle.

b. Name three pairs of supplementary angles.c. Name a pair of complementary angles.d. Give possible measures for ∠4 and ∠1. e. Determine m∠11.

review

draw and label a diagram to represent each statement. 25. ∠PAM is acute 26. ∠BIG is obtuse 27. ∠RHT is right

use a protractor to measure each angle to the nearest degree.

28. 29. 30.

tic-tAc-toe ~ Angle Art

step 1: Research how angles and lines are used in art. Write at least a one page summary of your fi ndings.

step 2: Create a work of art using the diff erent types of angles in Block 1. Use an 11 inch by 18 inch piece of paper for your work of art.

Give your creation a title and sign it.

1 24

3 5

1011

7 68 9

18 Lesson 4 ~ Vertical Angles

verticaL angLes

Lesson 4

Complementary and supplementary angles are types of special angles. Another pair of special angles are vertical angles. Vertical angles are formed by two intersecting lines. Th ey have a common vertex but are not adjacent.

Two adjacent angles whose non-common sides are opposite rays are a linear pair. In the diagram above, there are four sets of linear pairs. For example, ∠1 and ∠3 form a linear pair. If two angles form a linear pair, they are supplementary.

step 1: Trace ∠3 above onto a sheet of paper.

step 2: Place the traced ∠3 on top of ∠4. What do you notice?

step 3: Repeat steps 1 and 2 with ∠1 and ∠2. What do you fi nd?

step 4: Draw two intersecting lines on a piece of paper.

step 5: Label the angles that are formed with the numbers 5, 6, 7 and 8. Identify the vertical angles in your drawing.

step 6: Measure each angle in your drawing with a protractor.

step 7: What can you conclude about the measure of any pair of vertical angles?

expLOre! the verticaL angLe reLatiOnship

140°

140° 40°40°

22°

22°

158° 158°

|

|

|| ||

||

||

|

|

13

24

Lesson 4 ~ Vertical Angles 19

Sierra and King used different methods to find the solution to the question below. Look at Sierra’s and King’s work.

Question sierra’s Work King’s WorkIf m∠2 = 36° and m∠3 = 144°,what is the measure of ∠4?

Sierra knows that ∠4 and ∠3 are supplementary because they are a linear pair. She subtracted 144 from 180.

180° − 144° = 36° m∠4 = 36°

King knows that ∠2 and ∠4 are vertical angles. They have the same degree measure.

m∠4 = 36°

There is often more than one way to arrive at a correct answer. Both Sierra and King answered the question correctly but used different methods.

Find the measure of each missing angle. a. m∠3 b. m∠1 c. m∠4

a. Vertical angles are congruent. m∠2 = m∠3 54° = m∠3 b. ∠1 and ∠2 are a linear pair. m∠1 + m∠2 = 180° Substitute 54° for m∠2. m∠1 + 54° = 180° Subtract 54° from each side of the equation. m∠1= 126° c. Vertical angles are congruent. m∠1 = m∠4 126° = m∠4

In geometry, sketches are used as a visual representation for the information given. Sketches are not always accurate in terms of actual length or degree measure. Information given in a diagram should be used to solve a problem rather than measuring the actual lengths and degree measures with a ruler and protractor.

For example, the sketch below shows that the angle is 45° and the length of the segment is 5 feet. The angle in the sketch may not measure exactly 45° and the line does not measure 5 feet. You should still use this information to solve any problem asked about the diagram.

exampLe 1

solutions

45° 5 feet

43

2

13

4254°


use the diagram at right.a. solve for x. b. Find the measure of each angle.

a. Vertical angles have equal measures. 3x + 7 = x + 30 Subtract x from each side of the equation. 2x + 7 = 30 Subtract 7 from each side of the equation. −7 −7 2x = 23 Divide each side of the equation by 2. x = 11.5

b. Write the given expression (3x + 7)° (x + 30)° for each angle. Substitute the solution for x. (3(11.5) + 7)° (11.5 + 30)° Multiply. 34.5 + 7 Add. 41.5° 41.5° The measure of each angle is 41.5°.

Each special angle pair has properties that are important to remember. The sum of complementary angles is 90°. The sum of supplementary angles or linear pairs is 180°. Vertical angles are congruent.

exercises

use the diagram at right. determine the measure of each unknown angle. 1. If m∠1 = 50°, find the following: 2. If m∠4 = 153°, find the following: a. m∠2 = ? a. m∠1 = ? b. m∠3 = ? b. m∠2 = ? c. m∠4 = ? c. m∠3 = ? 3. If m∠3 = 126°, find the following: 4. If m∠2 = 16°, find the following: a. m∠1 = ? a. m∠1 = ? b. m∠2 = ? b. m∠3 = ? c. m∠4 = ? c. m∠4 = ?

sketch a diagram to represent each situation. Label each diagram.

5. ∠7 and ∠8 are vertical angles. 6. ∠9 and ∠10 are obtuse vertical angles.

7. ∠1 and ∠2 are a linear pair. 8. ∠ABC and ∠DBE are vertical ∠1 is an acute angle. and complementary angles.

exampLe 2

solutions

(3x + 7)°

(x + 30)°

13

24

Lesson 4 ~ Vertical Angles 21

Identify each special angle pair as vertical angles or a linear pair. solve for x. Check your solution. 9. 10. 11.

12. 13. 14.

15. 16. 17.

18. Use the diagram at the right. a. Solve for x. b. Find m∠ABC. c. Find m∠CBD.

19. Use the diagram at the right. a. Solve for x. b. Find m∠WXY. c. Find m∠TXY.

(6x + 4)° (4x − 14)°

(5x + 4)°

86°

(4x + 30)°

(x + 11)°(3x + 7)° 6x°

(3x + 10)°62°

(5x + 9)°

(8x + 3)° (7x + 12)°89°

(10x − 7)°(6x + 13)°

UX

T

W

Y

114°

(2x − 1)°

A

C

DB

(4x + 7)°

(8x + 2)°

42°


review

Match each diagram to a description from the word bank. some diagrams may match more than one description.

20. 21. 22.

23. m∠PQR = 110° 24.

|| 25.

m∠ABC = 70°

26. 27. 28.

Acute angle Obtuse angle Right angle

Congruent angles Straight angle Vertical angles

Supplementary angles Linear pair Complementary angles

P

Q

R

50° 130°

tic-tAc-toe ~ BooK oF Poe try

step 1: Write a separate poem about each special angle pair: linear pair, supplementary, complementary, vertical, alternate interior, alternate exterior, corresponding and same-side interior angles. Title each poem with the name

of the special angle pair. step 2: Put the eight poems in a booklet. Include diagrams and illustrations.

K

I S

||||

50°

40°

142°

Lesson 5 ~ Alternate Exterior And Interior Angles 23

aLternate exteriOr and interiOr angLes

Lesson 5

The words “interior” and “exterior” are used in many non-math situations. Th e word “interior” is used to describe things that are inside. Th ings that are exterior are on the outside. Th e two special angle pairs in this lesson are called alternate interior angles and alternate exterior angles. Look at the diagrams in the table below. What does it mean for a pair of angles to be alternate exterior angles? What about alternate interior angles?

A transversal is a line that intersects two or more lines. Alternate exterior angles are two angles on the outside of two lines and on opposite sides of a transversal.Alternate interior angles are two angles on the inside of two lines and on opposite sides of a transversal.

Alternate exterior Angles Alternate Interior Angles

∠6 and ∠8 are alternate exterior angles.




∠1 and ∠2 are alternate interior angles.




X

Y

Q

A

P

B

765

8

321

4>>

>>

In this diagram, ‹

___ › XY is the transversal.

‹ ___›

AB || ‹

___ › PQ reads, “Line AB is parallel to

line PQ.”

5 6

8 7

1819 16

17

1014

1213

3 1

2 4

>>

>>

>>

>>

24 Lesson 5 ~ Alternate Exterior And Interior Angles

step 1: Draw two parallel lines intersected by a transversal. Make the diagram large enough to easily measure the angles with a protractor. Label your angles 1 through 8.

step 2: Use a protractor to measure the eight angles on the drawing from step 1. Record their measures.

step 3: Which pairs of angles in your diagram are alternate exterior angles? What do you notice about the degree measure of each pair of alternate exterior angles?

step 4: Which pairs of angles are alternate interior angles? What do you notice about the degree measure of each pair of alternate interior angles?

step 5: Trace the fi gure from the bottom of page 23 onto a piece of paper.

step 6: Slide your tracing until angles 1, 2, 3 and 4 are on top of angles 5, 6, 7 and 8. a. What do you observe? Explain. b. Does this confi rm your conclusion in steps 3 and 4?

step 7: Draw two lines that are not parallel. Draw a transversal that intersects both lines.

step 8: Use a protractor to measure the eight angles created in step 7.

step 9: Write a conclusion about the alternate interior angles and alternate exterior angles formed by parallel lines compared to those formed by non-parallel lines.

name the angle relationship between ∠1 and ∠2. determine whether ∠1 and ∠2 are congruent.a.

1

2

b. 1

2

c. 1

2

a. Alternate interior. Th e angles are congruent because the lines are parallel.b. Alternate exterior. Th e angles are congruent because the lines are parallel.c. Alternate exterior. Th e lines are not parallel, so the angles are not congruent.

exampLe 1

solutions

expLOre! aLternate exteriOr and aLternate interiOr

<

< <<

<<


Identify the special angle pair relationship. solve for x.

The angles are alternate exterior angles.

Lines l and m are parallel so the alternate exterior angles are congruent. 8x + 4 = 86Subtract 4 from each side of the equation. −4 −4Divide each side of the equation by 8. 8x __ 8 = 82 __ 8

x = 10.25

☑Check the solution. (8x + 4)° = 86° Substitute 10.25 for x. (8(10.25) + 4)° ?= 86° Add. 82 + 4° ?= 86° 86° = 86°

use the figure at right. a. solve for x. b. Find the measure of the angles.

a. The lines are parallel so alternate interior angles are congruent. 9x − 58 = 2x + 5 Subtract 2x from each side of the equation. −2x −2x Add 58 to each side of the equation. 7x − 58 = 5 +58 +58 Divide each side of the equation by 7. 7x __ 7 = 63 __ 7 x = 9

b. Write the given expression for each angle. (9x − 58)° (2x + 5)° Substitute 9 for x found in part a. (9(9) − 58)° (2(9) + 5)° Multiply. (81 − 58)° (18 + 5)° Simplify. 23° 23° Each angle measures 23°. Alternate interior angles are congruent so part b verifies the solution for x.

exampLe 2

solution

exampLe 3

solutions

(8x + 4)°86°

l m

(9x − 58)°

(2x + 5)° >>

>>

<<<<


exercises

use one of the following special angle pairs to identify the relationship of the angles shown.

1. 2. 3.

4. 5. 6.

7. 8. 9.

name the special angle pair relationship between ∠1 and ∠2. explain whether ∠1 ≅ ∠2.

10. 11. 12.

name the special angle pair relationship. solve for x.

13. 14. 15.

1

2

90°

90°

Alternate exterior Alternate Interior Vertical Linear Pair

57°

57°

70°

(3x + 10)°>

> (4x + 3)°

103°>

>120°

8x°

>>>>1

2 >>

>>>>1

2

>>

65°65°

50°130°

y°

y°3

4

140° 40°

21

54

>>

>> >>

>>


solve for x. Th en fi nd the measure of each identifi ed angle. Check your solution. 16. 17. 18.

19. 20. ∠1 and ∠2 are vertical angles 21. ∠5 and ∠8 are m∠1 = (5x + 7)° supplementary m∠2 = (3x + 15)° m∠5 = (3x − 40)° m∠8 = (7x − 120)°

22. Explain how to distinguish between alternate exterior angles and alternate interior angles.

review

Write two possible names for each angle. 23. 24. 25.

sketch and label a diagram for each description. 26. an acute angle 27. vertical angles with each angle measuring 40°

28. adjacent supplementary angles 29. complementary angles that are not adjacent

2x°

56°

tic-tAc-toe ~ tr A nsV e rs A l col l Age

step 1: Find at least 10 photographs or pictures of transversals. You can take photographs, locate and print pictures from the internet or cut pictures from magazines or newspapers.

step 2: Identify the transversal and special angle pairs in each picture.

step 3: Make a collage to display the pictures.

(6x − 7)°(5x + 10)°

>>

>>

(x + 100)°

(6x + 20)°

>>

>>(3x − 22)°

(2x + 12)°

>>

>>

K

5

A

B

M

3P

o

W


tic-tAc-toe ~ Bisecti ng Angle s

To bisect an angle means to cut it in half. You can bisect an angle using two diff erent methods. One construction method to bisect an angle uses patty paper. Another method uses a compass and straightedge.

Bisecting an Angle using Patty Paper or tracing Paper

step 1: Draw or trace an angle onto a piece of patty paper or tracing paper. Record the measure of the angle.

step 2: Fold one ray of the angle onto the other ray of the angle.

step 3: Trace the crease and measure each angle.

Bisecting an Angle using A Compass and straightedge

step 1: Draw or trace an angle. Record the measure of the angle.

step 2: Place the stylus or sharp point of a compass on the vertex. Use the compass to draw an arc through the angle.

step 3: Place the stylus on one of the points of intersection between a ray of the angle and the arc from step 2. Draw an arc as shown.

step 4: Repeat step 3 at the other point of intersection.

step 5: Use a straightedge to draw a ray from the vertex to the intersection of the two arcs drawn in steps 3 and 4.

step 6: Measure each angle.

1. Use a protractor and patty/tracing paper to draw 90°, 24°, 115° and 160° angles. Each angle should be on a separate sheet of patty/tracing paper.

2. Bisect each angle by folding.3. Use a protractor to draw another set of angles with the measures listed in # 1 on regular paper.4. Bisect each angle using a compass and straightedge.5. Lay your matching patty/tracing paper constructions on top of each compass construction.6. Summarize each method of construction. Include which method you prefer and explain why.

Discuss the pros and cons of each method.

Place stylus here.

Lesson 6 ~ Corresponding And Same - Side Interior Angles 29

Two additional special pairs of angles formed by two lines and a transversal are corresponding angles and same-side interior angles.

Corresponding angles are two angles on the same side of a transversal. One angle is an exterior angle and the other angle an interior angle. Th ey are not adjacent angles.

Same-side interior angles are between the two lines on the same side of a transversal.

step 1: If two parallel lines are cut by a transversal, predict: a. What is the relationship between a pair of same-side interior angles? b. What is the relationship between a pair of corresponding angles?

step 2: Use a straight edge to draw a pair of parallel lines with a transversal. Make the fi gure large enough that you can measure the angles with a protractor.

step 3: Use a protractor to measure each of the eight angles. Record each measure.

step 4: What do you observe about the corresponding angle measures?

step 5: Look at the measures of the same-side interior angles. What do you observe about their measures?

step 6: Must the lines be parallel for these relationships to occur? Draw a fi gure without parallel lines to test your answer.

786 5

4321

expLOre! mOre angLe pairs

cOrrespOnding and same - side interiOr angLes

Lesson 6

30 Lesson 6 ~ Corresponding And Same - Side Interior Angles

name the special angle pair relationship between ∠1 and ∠2.a. b. c.

a. corresponding anglesb. same-side interior anglesc. corresponding angles

Claudia and Ping come to diff erent conclusions when fi nding the missing angles in the diagram below. Determine who is correct and why.

Two lines in the diagram are intersected by a transversal. It is not known whether the lines are parallel. Unless lines are marked parallel it cannot be assumed they are parallel. Ping is correct. Th e measures of ∠3, ∠4, ∠5 and ∠6 cannot be determined.

exampLe 1

solutions

1

2

786 5

4321

Claudia’s Work Ping’s Workm∠1 = 72°m∠2 = 108°m∠3 = 108°m∠4 = 72°m∠5 = 72°m∠6 = 108°m∠7 = 72°

m∠1 = 72°m∠2 = 108°m∠3 = cannot determinem∠4 = cannot determinem∠5 = cannot determinem∠6 = cannot determinem∠7 = 72°

>>

>>

7

1 2 3

5

4

6108°

12

12


Write an equation and solve for x.

Corresponding angles have equal measures. 4x + 9 = 117 Subtract 9 from each side of the equation. −9 −9 Divide each side of the equation by 4. 4x __ 4 = 108 ___ 4

x = 27

☑ Check the solution. Substitute 27 for x in the equation. 4(27) + 9 ?= 117 108 + 9 ?= 117 117 = 117

Write an equation and solve for x. Then find the measure of each identified angle.

The lines are parallel so the same sideinterior angles sum to 180°. (3x + 1) + (3x + 44) = 180 Combine like terms. 6x + 45 = 180 Subtract 45 from each side of the equation. −45 −45Divide each side of the equation by 6. 6x __ 6 = 135 ___ 6 x = 22.5

Find the measure of each angle. (3x + 1)° (3x + 44)° Substitute 22.5 for x. (3(22.5) + 1)° (3(22.5) + 44)° (67.5 + 1)° (67.5 + 44)° 68.5° 111.5° ☑68.5° + 111.5° ?= 180° 180° = 180°

exampLe 2

solution

(4x + 9)°

117°

exampLe 3

solution

>>

>>

(3x + 1)°

(3x + 44)°<<

<<


exercises

name the special angle pair relationship between ∠1 and ∠2 in each diagram.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. Which diagrams in exercises 1-9 have parallel lines?

solve for x. 11. 12. 13.

solve for x. Then find the measure of each identified angle. Check your solution.

14. 15. 16.

21

2

1

2

1

2

1

<<

<<

110°

(x + 6)°

>>

>> 46°

(4x + 40)°

>>

>>

(62 − 2x)°

>>>>

(90 − 5x)°

>>

21

<<

<<2

1>

>

1

22

1<<

<<

2

1

<<

<<

4x°

44°

>>

(47 + 2x)°

129°>>

>>

(5x + 34)°

154°>>


Find the measure of each numbered angle. Check your solution.

17. 18. 19.

sketch a diagram to represent each special pair of angles. Label the angles in each pair as ∠1 and ∠2. 20. congruent same-side interior angles 21. alternate interior angles that are not congruent

22. acute corresponding angles 23. right alternate exterior angles

24. obtuse same-side interior angles 25. acute vertical angles

review

use a protractor and/or straightedge to draw each diagram. Label each diagram. 26. m∠GUM = 162° 27. m∠RPQ = 20°

28. 3 parallel lines with transversal ‹

___ › AB 29. right angle named ∠2

name each special angle pair. solve for x.

30. 31. 32.

>>

>>

81°132

4 56 7

>>

>> 1

3

2

4(8x + 25)°

(10x + 5)°

12

34

56

(4x + 2)°

(6x − 7)°

119°x°2x°

3x°(5x + 1)°

(3x + 44)°

tic-tAc-toe ~ Fl i P BooK

step 1: Create a fl ip book showing all of the special angle pairs in Block 1. ◆ Supplementary angles ◆ Vertical angles ◆ Corresponding angles ◆ Alternate interior angles ◆ Same-side interior angles ◆ Alternate exterior angles ◆ Complementary angles ◆ Linear pairs

step 2: Include a defi nition, a diagram and an example for each special angle pair.

Special Angle Pairs


tic-tAc-toe ~ cros swor d

Create a crossword puzzle using all of the vocabulary from Block 1. Make a blank master copy and an answer key.

tic-tAc-toe ~ PuZZl i ng Angle s

Find the measure of the numbered angles in each diagram.You will need to research about the sum of the angles in a triangle before completing the puzzle.

40413938

67

568°

810

911

1615

3735

36

17

12

18

19

3332

3431

23

21

2429 2526

2227

3028

20

120°

14

1314

32>>

>>

>>

Block 1 ~ Review 35

Lesson 1 ~ Measuring and Naming Angles

use a protractor to measure each angle to the nearest degree. 1. 2. 3.

List the four names for each angle.

4. 5.

sketch and label a diagram to represent each angle. use a protractor, as needed. 6. m∠YOU = 125° 7. m∠BAT = 40° 8. ∠HAM and ∠HAP are adjacent

O

M P

5Y5

Z

X

review

acute angle corresponding angles right angle adjacent angles degree same-side interior angles alternate exterior angles linear pair straight angle alternate interior angles obtuse angle supplementary angles angle protractor transversal complementary angles ray vertex congruent vertical angles

vocabulary

BLoCK 1

36 Block 1 ~ Review

Lesson 2 ~ Classifying Angles

Classify each angle as acute, right, obtuse or straight.

9. 10. 11.

sketch and label a diagram.

12. ∠CAT ≅ ∠DOG 13. a right angle that can be named three different ways

set up an equation and solve for x. Check your solution.

14. 15. 16.

17. m∠BIG = (x + 14)° a. What must x be equal to if ∠BIG is a right angle? b. What must x be less than if ∠BIG is acute? c. What must x be greater than if ∠BIG is obtuse?

Lesson 3 ~ Complementary and Supplementary Angles

Identify each pair of angles as complementary, supplementary or neither.

18. 19. 20.

solve for x. Check your solution.

21. 22. m∠A = (25 + 2x)° 23. m∠P = (10 + 3x)° ∠A and ∠P are complementary.

(3x – 12)°

(35 + 2x )°31°

60° 120°

135°

40°

51°

39°

(x + 7)°

42° 142° x°

51°134°

(2x + 16 )°

Block 1 ~ Review 37

sketch a diagram for each situation. Label it and solve for x.

24. ∠PAR and ∠TYE are supplementary; m∠PAR is 83° and m∠TYE is (x + 5)°

25. ∠F and ∠G are supplementary; m∠F is 46° and m∠G is (3x – 25)°

26. ∠1 and ∠2 are complementary; m∠1 is 2x° and m∠2 is 3x°

Lesson 4 ~ Vertical Angles

determine the measure of the angles labeled a, b and c.

27. 28.

name the special angle pair. solve for x. Check your solution. 29. 30. 31.

32. 33. 34.

Lesson 5 ~ Alternate Exterior and Interior Angles

name the special angle pair relationship between ∠1 and ∠2. explain whether or not the angles are congruent.

35. 36. 37.

a

c45°b

b

123°ca

(x + 8)°

141°(2x + 12)°

(5x + 27)° (6x + 7)°

(4x + 10)° (5x)°

(5 + 3x)°(10 + 7x)° (x + 76)°

(5x)°

1

2 2

1>>

>>

>>

>>

2

1

38 Block 1 ~ Review

solve for x. Then find the measure of each angle. Check your solution.

38. 39. 40.

Lesson 6 ~ Corresponding and Same-Side Interior Angles

name the special angle pair relationship between ∠1 and ∠2.

41. 42. 43.

44. Copy the table below. List each special angle pair from the blue box in one category. Assume lines are parallel for *angle pairs.

Linear Pairs Vertical Angles*Alternate exterior Angles Complementary Angles*Alternate Interior Angles *same-side Interior Angles*Corresponding Angles supplementary Angles

have a sum of 90° have a sum of 180° Are equal in measure

Write an equation to solve for x. Then find the measure of each angle. Check your solution.

45. 46. 47.

>>

>>

(3x − 35)°

(x + 11)°>>

>>(81 + x)°(60 − 2x)°

>>

>>

(5x + 2)°(3x + 40)°

2

1

2

1

>>

>>

>>

>>

2

1

(2x + 10)°

>>

>> (x + 5)°

(3x + 70)°>>

>> (7x + 30)°(8x − 12)°

>>

>>

(48 − 2x)°

Block 1 ~ Review 39

Jerry nursery mAnAger

turner, oregon

I am a nursery manager. I work with a staff planning and planting 25 different tree species. Some of the seedlings I plant are very recognizable. I plant Douglas Fir, Noble Fir, Ponderosa Pine and Giant Sequoia. Some of the trees I grow are for planting around homes. Others go into the woods for reforestation. When loggers cut down trees, the trees I grow are planted in their place. This helps make sure that there will be trees to harvest in the future. I help grow 6.5 to 7.5 million trees for about 150 different customers. I work at a small nursery, but some nurseries in the Northwest grow over 25 million trees a year.

I use math every day in my job. When preparing to plant, I have to know how many trees will fit on each acre. Some seedlings are planted at 75 per foot and some are planted at 18 per foot. Math helps me to determine how many seeds I will need to make sure I will have enough trees to fill a customer’s order. Fertilizers are another area where I use math. I take soil samples and calculate at what rate I should apply fertilizer to make sure that the seedlings grow well. When I prepare to ship trees to a customer, I also have to use math to make sure that they are counted correctly and packaged right for each customer.

Most nursery managers have a Bachelor of Science degree in Agriculture or a related horticultural field. These degrees require high-level math classes like calculus. About 10% of nursery managers do not have a college degree. Those managers usually work their way up to a manager position with many years of work and experience. Nursery manager salaries start at around $35,000 to $40,000 per year and can get as high as $65,000 or more per year. Often the pay is related to the size of the nursery and how many trees the nursery sells.

I feel lucky to be doing something I enjoy and making a positive contribution to the world I live in by growing trees. I have helped grow over 180,000,000 trees in my career. I enjoy going to the woods and seeing stands of trees I grew. I know that some of those trees will still be living when my grandchildren’s grandchildren are old.

CAreer FoCus

block 1 ~ lines and angles - oregon focusblock 1 ~ lines and angles ~ angle pairs 1 block 1 ~ lines...

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