1.4 Angles and Their Relationships 37

Reasoning from the definition of an angle bisector, the Angle-Addition Postulate,and the Protractor Postulate, we can justify the following theorem.

This theorem is often stated, "The angle bisector of an angle is unique." This statementis proved in Example 5 of Section 2.2.

::<der:.\hy::-:ate.

1. What type of angle is each of the following?a) 4]0 b) 90° c) 137.3°

2. What type of angle is each of the following?a) 115° b) 180° c) 36°

3. What relationship, if any, exists between two angles:a) with measures of 37° and 53°?b) with measures of 37° and 143°?

4. What relationship, if any, exists between two angles:a) with equal measures?b) that have the same vertex and a common side

between them?

In Exercises 5 to 8, describe in one word the relationshipbetween the angles.

5. LABD and LDBC 6. L7 and L8

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7 58 i6 m

7. Ll and L2 8. L3 and L4

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H

3 4~,------~,~----~E F GB C

Use drawings as needed to answer each of the followingquestions.

9. Must two rays with a common endpoint be coplanar?Must three rays with a common endpoint be coplanar?

------*~~-» ------1-1O. Suppose that AB, AC, AD, AE, and AF are coplanar.

Exercises 70- 73

Classify the following as true or false:a) mLBAC +mLCAD = mLBADb) LBAC ~ LCADc) mLBAE - mLDAE = mLBACd) LBAC and LDAE are adjacente) mLBAC +mLCAD +mLDAE = mLBAE

11. Without using a protractor, name the type of anglerepresented by:a) LBAE b) LFAD c) LBAC d) LFAE

12. What, if anything, is wrong with the claimmLFAB + mLBAE= mLFAE?

13. LFAC and LCAD are adjacent and Aft and AD are oppositerays. What can you conclude about LFAC and LCAD?

For Exercises 14 and 15, let mL1 = x and mL2 = y.

14. Using variables x and y, write an equation that expressesthe fact that L I and L2 are:a) supplementary b) congruent

15. Using variables x and y, write an equation that expressesthe fact that L I and L2 are:a) complementary b) vertical

For Exercises 16,17, see figure on page 38.

16. Given: mLRST=39°mLTSV=23°

Find: mLRSV17. Given: mLRSV=59°

mLTSV= 17°Find: mLRST

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38 CHAPTER 1 tJ LINE AND ANGLE RELATIONSHIPS

18. Given: mLRST=2x+9 /?mLTSV=3x-2mLRSV=67°

s~Find: x19. Given: mLRST=2x-10

mLTSV=x+6mLRSV = 4(x - 6) Exercises 16-24

Find: x and mLRSV20. Given: mLRST= 5(x+ 1) - 3

mLTSV = 4(x - 2) + 3mLRSV= 4(2x + 3) - 7

Find: x and mLRSV

,,\ 21. Given: mLRST=~\ mLTSV=~

mLRSV=45°Find: x and mLRST

22. Given: mLRST=~ 3mLTSV=~mLRSV=49°

Find: x and mLTSV--->

23. Given: ST bisects LRSVmLRST=x+ymLTSV=2x-2ymLRSV=64°

Find: x andy--->

24. Given: ST bisects LRSVmLRST = 2x + 3ym L TSV = 3x - y + 2mLRSV = 80°

Find: x andy<----> +-----> <---->

25. Given: AB and AC in plane P as shown; AD intersectsPat point ALCAB = LDACLDAC= LDABWhat can you conclude?

(=~~/ ~;/~L-7~--26. Two angles are complementary. One angle is 12° larger

than the other. Using two variables x and y, find the sizeof each angle by solving a system of equations.

27. Two angles are supplementary. One angle is 24° morethan twice the other. Using two variables x and y, find themeasure of each angle.

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28. For two complementary angles, find an expression for themeasure of the second angle if the measure of the first is:a) XO

b) (3x - 12tc) (2x + 5yt

29. Suppose that two angles are supplementary. Findexpressions for the supplements, using the expressionsprovided in Exercise 28, parts (a) to (c).

30. On the protractor shown, NP bisects LMNQ. Find x.

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Exercises 3D, 31

31. On the protractor shown for Exercise 30, LMNP andLPNQ are complementary. Find x.

32. Classify as true or false:a) If points P and Q lie in the interior of LABC, then PQ

lies in the interior of LABC.b) If points P and Q lie in the interior of LABC, then PQ

lies in the interior of LABC.c) If points P and Q lie in the interior of LABC, then PQ

lies in the interior of LABC.

In Exercises 33 to 40, use only a compass and a straightedgeto perform the indicated constructions.

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Exercises 33-35

33. Given:Construct:

34. Given:Construct:

35. Given:Construct:

Obtuse LMRPWith oA as one side, an angle = LMRPObtuse LMRPRS, the angle bisector of LMRPObtuse LMRPRays !?S, Iff, and RfJ so that LMRP isdivided into four = anglesStraight LDEFA right angle with vertex at E

36. Given:.Construct:

(HINT: Use Construction 4.)

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1.5 Introduction to Geometric Proof 39

37. Draw a triangle with three acute angles. Construct anglebisectors for each of the three angles. On the basis of theappearance of your construction, what seems to be true?

38. Given: Acute L 1 and ABConstruct: Triangle ABC with LA "" L I, LB "" L 1,

and side AB

:"42. IfmLTSV= 38°, mLUSW=40°, and mLTSW= 61°,find mLUSV.

u

~-----.A B L v

IN

Exercises 42, 4339. What seems to be true of two of the sides in the triangle

you constructed in Exercise 38?Given: Straight LABC and EDConstruct: Bisectors of LABD and LDBCWhat type of angle is formed by the bisectors of the twoangles?

43. IfmLTSU = x + 2z, mLUSV = x - z, andmL VSW = 2x - z. find x if mLTSW = 60,Also, find z if mL USW = 3x - 6.Refer to the circle with center P.a) Use a protractor to find mL 1.b) Use a protractor to find m L 2,c) Compare results in parts (a) and (b).

"40.

44.

!f-~+-----"J v41. Refer to the circle with center O.

a) Use a protractor to find mLB,b) Use a protractor to find mLD.c) Compare results in parts (a) and (b).

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8

A I?------<>---~ C

, 45. On the hanging sign, the three angles (LABD, LABC ,and LDBC) at vertex B have the sum of measures 360°, IfmLDBC = 90° and SA bisects the indicated reflex angle,find mLABC.

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ProofAlgebraic Properties

Given Problem and ProveStatement

Sample Proofs

To believe certain geometric principles, it is necessary to have proof. This section in-troduces some guidelines for proving geometric properties. Several examples are of-fered to help you develop your own proofs. In the beginning, the form of proof will bea two-column proof, with statements in the left column and reasons in the right column.But where do the statements and reasons come from?

i •••• :•••.:.2.: •• :1 •••••••••••••••••••••••••••••••••••••••••••••