Fahrenheit =Lrr Celsius UNIT2 Angles and their Measurements

UNIT2 Angles and their Measurements

ANGLE A figure formed by two rays that have the same endpoint

Sides of an Angle: , two rays

Vertex of an Angle: endpoint of the rays

Can be called: MEASURING ANGLES: Using a Protractor (THINK ROTATION….) Protractor Postulate Angle Measurement = |a – b|

=________ =__________ =__________ =___________|140–45|=95 |45–0|=45 |180–0|=180 |0–0|=0

∠L-3 AnglesAn angle ( L) is a figure formed by two rays that have the sameendpoint. The two rays are called the sides of the angle, andtheir common endpoint is the vertex of the anBle

The sides of the angle shown are BA and BC. The vertex ispointB. Theanglecanbe called LB, IABC, LCBA,or Ll. lfthree letters are used to name an angle, the middle lettcr-namesthe vertex.

When there is no possibility of confusion, an angle can benamed by just its vertex, as LB above. On the other han(, itwould be incorrect to refer to /- Q in the diagram at the ri!,ht,since point Q is the vertex of three angles: l RQT, L RQS, andlSQT.

l0 / Chaprer I

G H M N T Y Z

-4-2 0 2 4 6 8

Exs.41-4E

44. lxl>0 as.@l<a

Name the graph of the given equation or inequality.

ExampleSolution

a, x)2a. NT

b.

b.

4<.x1.6TY

b. A11 points in MT that are not in N7

41. -2 1x 12 42. x 10 43. lxl : 0

Write an inequality whose graph is described.

Example u. nGSolution a.x1-246. M2

b.0<x1247. HZ

49. The Ruler Postulate suggests that there are many ways to assign coordi-nates to a line. The Fahrenheit and Celsius temperature celsiusscales on a thermometer indicate two such ways of assigningcoordinates. A Fahrenheit temperature of 32" correspondsto a Celsius temperature of 0'. The formula, or rule, forconverting a Fahrenheit temperature F into a Celsius tem-perature C is

c =Lrr -32\.9'a. What Celsius temperatures correspond to Fahrenheit

temperatures of 98.6' and -40"? ob. Solve the equation above for ,F to obtain a rule for con-

verting Celsius temperatures to Fahrenheit temperatures.c. What Fahrenheit temperatures correspond to Celsius

temperatures of 100" and -10"?

+48. All points in MY that are not in NG

Fahrenheit

AB! "!!

BC! "!!

B

L-3 AnglesAn angle ( L) is a figure formed by two rays that have the sameendpoint. The two rays are called the sides of the angle, andtheir common endpoint is the vertex of the anBle

The sides of the angle shown are BA and BC. The vertex ispointB. Theanglecanbe called LB, IABC, LCBA,or Ll. lfthree letters are used to name an angle, the middle lettcr-namesthe vertex.

When there is no possibility of confusion, an angle can benamed by just its vertex, as LB above. On the other han(, itwould be incorrect to refer to /- Q in the diagram at the ri!,ht,since point Q is the vertex of three angles: l RQT, L RQS, andlSQT.

l0 / Chaprer I

G H M N T Y Z

-4-2 0 2 4 6 8

Exs.41-4E

44. lxl>0 as.@l<a

Name the graph of the given equation or inequality.

ExampleSolution

a, x)2a. NT

b.

b.

4<.x1.6TY

b. A11 points in MT that are not in N7

41. -2 1x 12 42. x 10 43. lxl : 0

Write an inequality whose graph is described.

Example u. nGSolution a.x1-246. M2

b.0<x1247. HZ

49. The Ruler Postulate suggests that there are many ways to assign coordi-nates to a line. The Fahrenheit and Celsius temperature celsiusscales on a thermometer indicate two such ways of assigningcoordinates. A Fahrenheit temperature of 32" correspondsto a Celsius temperature of 0'. The formula, or rule, forconverting a Fahrenheit temperature F into a Celsius tem-perature C is

c =Lrr -32\.9'a. What Celsius temperatures correspond to Fahrenheit

temperatures of 98.6' and -40"? ob. Solve the equation above for ,F to obtain a rule for con-

verting Celsius temperatures to Fahrenheit temperatures.c. What Fahrenheit temperatures correspond to Celsius

temperatures of 100" and -10"?

+48. All points in MY that are not in NG

Fahrenheit

You can use a protractor like the one shown below to find the measure indegrees of an angle. Although angles are sometimes measured in other units,you may assume that any angle measure in this book is in degrees.

Using the outer (red) scale of the protractor, you can see that the (degree)measure of LXOY is 40. You can indicate this by writing

mLXOY = 40.

dl .io so eo lw t tb ;-/)rs S.r*to,oo eo u a"'2o,

,14t" - "o.

With just one placement of a protractor, you can find the measures ofseveral angles that have a common vertex. Using the inner scale of the protrac-tor, you frnd that:

wLYOZ - 140 ru/-WOZ - 45 wLYOW - 140 - 45 - 95

Using a protractor involves the following basic assumption.

hsob

oN

-8r$o

Postulate 3 Protractor PostulateOn ii in a given plane, choose any point O between A and3. Consider O7

.+-€and OE and all the ravs that can be drawn from 0 on one side of lB. Theserays can be paired wiitr ttr" real numbers from 0 to 180 in such a way that:

n. OZ is paired with 0, anA Oi with 180.

b. lf Oi is paired with r, and OQ with y, then m LPOQ = lx - yl.

Points, Lines, Planes, and Angles / ll

m∠YOW m∠WOZ m∠ZOX m∠ZOZ

ANGLES ARE CLASSIFIED BY THEIR MEASURES (degrees)

Reflex angle: Measure between 180 and 360 Zero angle: Measure 0

Angles are classified according to their measures.

Acute angle: Measure between 0 and 90Right angle: Measure 90Obtuse angle: Measure between 90 and 180Straight angle: Measure 180

LNOP is a straight angle.

Another basic assumption we make about angles is the following.

The small square indicates aright angle (rt. /-).

I"//B

./ ::\ \8\

\_AOC

<+AC,

Postulate 4 Angle Add,ition PostulateIf point B lies in the interior of LAOC, then

WLAOB*m/-BOC=mLAOC.

lf LAOC is a straight angle and .B is any point not onthen

WLAOBtmLBOC=180.

Congruent angles are angles that have equal measures.Since L DEF and /-GEH each have measure 45, you canwrite either

mLDEF - mLGEH or /-DEF: /-GEH.

Adjacent angles (adj. A) are two angles in a plane that have a comrnoh verdexand a common side but no common interior points.

/-3 and /4 are not adjacent angles.

VYfu/-l and /-2 are adjacent angles.

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ANGLE ADDITION POSTULATE

CONGRUENT ANGLES Angles that have the equal measures

ADJACENT ANGLES Two angles in a plane that have a common vertex and a common

side but no common interior points

Adjacent angles have shared rays EXTERIOR SIDES VS. INTERIOR SIDE OF TWO ADJACENT ANGLES

Exterior Side; Interior Side; ANGLE BISECTOR A ray that divides the angle in two congruent adjacent angles.

In the diagram . Therefore,






I"//B

./ ::\ \8\

\_AOC

<+AC,


WLAOB*m/-BOC=mLAOC.


WLAOBtmLBOC=180.











I"//B

./ ::\ \8\

\_AOC

<+AC,


WLAOB*m/-BOC=mLAOC.


WLAOBtmLBOC=180.











I"//B

./ ::\ \8\

\_AOC

<+AC,


WLAOB*m/-BOC=mLAOC.


WLAOBtmLBOC=180.











I"//B

./ ::\ \8\

\_AOC

<+AC,


WLAOB*m/-BOC=mLAOC.


WLAOBtmLBOC=180.






Perpendicular lines (I lines) are two lines that form right angles. Thisdefinition can be used in the following situations.

<+ € <+ <+l. If AB is perpendicular to CD (AB L CD), then each numberedangle is a right angle.

2. If any of the numbered angles is a right angle, then iE t t6.The word "perpendicular" is also used for intersecting rays and

segments that are parts of perpendicular lines. For example, ifdi ttrE in the diagram, then we can also say that CD L AB.

The following theorems are easily deduced by using the definition of per-pendicular lines. You will complete the proofs of the theorems in ClassroomExercise l0 and Written Exercises 9 and 10.

Theorem 7-4Adjacent angles formed by perpendicular lines are congruent.

Theorem 7-5.If two lines frirm congruent adjacent angles, then the lines are perpendicular.

Theorem 7-6If the exterior sides of two adiacent acute angles areangles are complementary.

Given: Ei t EiProve: LABC arrd LCBD are comp. l-.

perpendicular, then the

lnlrt7B-

The following example shows how these theorems can be used in a proof,

ErampleGiven: /-l: L2Prove: L3 and L4 are comp. 4-.

Proof:

Statements

Ll: L2kLt

/-3 and L4 are comp. l-.

GivenIf 2 lines form =, adj. A, then theare I.If the ext. sides of 2 adj. A are I, thenthe A are comp.

Reasons

3.

36 / Chapter I

BD! "!!

BA! "!!

BC! "!!

The bisector of an angle is a ray that divides the angle into twocongruent adjacent angles. In the diagram, mLXYW.:mLWYZand thus YWbisects /-XYZ.

Classroom Exercises1. What is the vertex of 14?2. Name the sides of 14.3. Name all angles adjacent to /-6.

State another name for the angle.

4. LACD 5. LABD 6. LEDC7. /_6 8. /_3 9. 15

f0. Why is it confusing to refer to /- B?11. Name three angles that have B as the vertex. Exs' 1-21

12. How many angles shown in the diagram have D as the vertex?

State whether the angle appears to be acute, right, obtuse, or straight.

13. lt16. LCDB

22. LBOC24. LFOG26. LBOG

14. L3I7. LADC

23, LHOG25. LFOC27. /_HOA

Complete.

19.m17 +mL6=mL ?

20. m16 + mL5 : ml 1

21. lf Di bisects /-CDA,then /- 1 : L 't

State the measure of each angle.

15. LEDB18. LADE

Exs.22-31

28. Name four angles that are adjacent to LFOG.29. Name some angles that are not adjacent to

' IFOG.30. What ray bisects an angle? Which angle(s)?31. Name a pair of:

a. congruent acute anglesb. congruent right anglesc. congruent obtuse angles

Points, Lines, Planes, and Angles / 13




4. LACD 5. LABD 6. LEDC7. /_6 8. /_3 9. 15




13. lt16. LCDB


14. L3I7. LADC


Complete.

19.m17 +mL6=mL ?

20. m16 + mL5 : ml 1



15. LEDB18. LADE

Exs.22-31








4. LACD 5. LABD 6. LEDC7. /_6 8. /_3 9. 15




13. lt16. LCDB


14. L3I7. LADC


Complete.

19.m17 +mL6=mL ?

20. m16 + mL5 : ml 1



15. LEDB18. LADE

Exs.22-31





ANGLES CONSTRUCTIONS www.mathopenref.com CONGRUENT ANGLES

PRACTICE

Construction 2Given an angle, construct an angle eongruent to the given angle.

Given: LABCConstruct: An angle congruent to /ABCProcedure:1. Draw a ray. Label it R?.2. Using,B as center and any radius, draw an arc iatersect-

ing Ei and, Et. Label the points of intersection Dand E.Using R as center and the same radius as before, drawan arc intersecting n7. mU"t the arc fr, with S thepoint where the arc intersects R7.Using S as center and a radius equal to DE, draw an arcthat intersects .8 at a point Q.

4.

5. Draw Rp.LQRS is congruent ro /-ABC.

Justification: lf DE and QS are drawn, 1'DBE -. AOR,S (SSS postulate).Thet lQRS:. /_ABC.

, t'{,

Construction 3Given an angle, construct the bisector of the angle.

Given: /-ABCConstruct: The bisector of lABCProcedure:l. Using B as center and any radius, draw an arc that inter-

sects ,B,4 at X arrd BC at Y.

2. Using X as center and a, suitable radius, draw an arc.Using Y as center and the same radius, draw an arc thatintersects the first arc at Z.

-3. Draw BZ.BZ bisects /-ABC.

Justification: DrawThus

LXIZ: LYBZ (SSS Postulate).BZ bisects /-ABC.

XZ and YZ. ThenLXBZ: LYBZ and

336 / Chapter 8

Exam,ple Given /-l and L2, constrtct an angle whosemeasure is equal to mll + mL2.

Solution First use Construction 2 to construct LLON con-gruent to Ll. Then use the same method to con-struct 1 MOL congrtentto 12 (as shown) so thatwLMON=m/-l+mL2.

In construction exercises, you won't ordinarily have to write out the proce-dure and the justif,rcation. However, you should be able to supply them whenasked to do so.

4*"KClassroom Exercises1. Given: LJKM

Explain how to construct a triangle that iscongruent to LJKM.

2. Draw any AB.a. Construct XJ so that XY - AB.b. Using X and I as centers, and a radius equal to AB, draw arcs that

intersect. Label the point of intersection Z.c. Draw XZ and lZ.d. What kind of triangle is LXYZ?

3. Explain how you could construct a 30' angle.4. Exercise 3 suggests that you could construct other angles with certain meas-

ures. Name some.

M

5. Suppose you are given the three lengths shown and areasked to construct a triangle whose sides have lengths r,s, and t. Can you do so? State the theorem from Chap-ter 4 that applies.

6. /-l and L2 are given. You see two attempts at constructing an angle whosemeasure is equal to mll + mL2. Are both constructions satisfactory?

JK

rF------{

L4OUI=mLl+mL2mLSAY-mLl+m/-2

Constructions and Loci / 337

ANGLE BISECTOR

Construction 2Given an angle, construct an angle eongruent to the given angle.

Given: LABCConstruct: An angle congruent to /ABCProcedure:1. Draw a ray. Label it R?.2. Using,B as center and any radius, draw an arc iatersect-

ing Ei and, Et. Label the points of intersection Dand E.Using R as center and the same radius as before, drawan arc intersecting n7. mU"t the arc fr, with S thepoint where the arc intersects R7.Using S as center and a radius equal to DE, draw an arcthat intersects .8 at a point Q.

4.

5. Draw Rp.LQRS is congruent ro /-ABC.

Justification: lf DE and QS are drawn, 1'DBE -. AOR,S (SSS postulate).Thet lQRS:. /_ABC.

, t'{,

Construction 3Given an angle, construct the bisector of the angle.

Given: /-ABCConstruct: The bisector of lABCProcedure:l. Using B as center and any radius, draw an arc that inter-

sects ,B,4 at X arrd BC at Y.

2. Using X as center and a, suitable radius, draw an arc.Using Y as center and the same radius, draw an arc thatintersects the first arc at Z.

-3. Draw BZ.BZ bisects /-ABC.

Justification: DrawThus

LXIZ: LYBZ (SSS Postulate).BZ bisects /-ABC.

XZ and YZ. ThenLXBZ: LYBZ and

336 / Chapter 8

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Fahrenheit =Lrr Celsius UNIT2 Angles and their Measurements