Download Angles and TheirRelationships 37

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 statement
is proved in Example 5 of Section 2.2.
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1O. Suppose that AB, AC, AD, AE, and AF are coplanar.
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?
Exercises 70- 73
Classify the following as true or false:
a) mLBAC + mLCAD = mLBAD
b) LBAC ~ LCAD
c) mLBAE - mLDAE = mLBAC
d) LBAC and LDAE are adjacent
e) mLBAC + mLCAD + mLDAE = mLBAE
In Exercises 5 to 8, describe in one word the relationship
between the angles.
5. LABD and LDBC
C
6. L7 and L8
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11. Without using a protractor, name the type of angle
represented by:
a) LBAE
b) LFAD
c) LBAC
d) LFAE
12. What, if anything, is wrong with the claim
mLFAB + mLBAE= mLFAE?
AD
Aft and
are opposite
rays. What can you conclude about LFAC and LCAD?
13. LFAC and LCAD are adjacent and
8. L3 and L4
7. Ll and L2
For Exercises 14 and 15, let mL1 = x and mL2 = y.
H
3 /
4
~,------~,~----~
B
C
E
F
G
Use drawings as needed to answer each of the following
questions.
9. Must two rays with a common endpoint be coplanar?
Must three rays with a common endpoint be coplanar?
14. Using variables x and y, write an equation that expresses
the fact that L I and L2 are:
a) supplementary
b) congruent
15. Using variables x and y, write an equation that expresses
the fact that L I and L2 are:
a) complementary
b) vertical
For Exercises 16,17, see figure on page 38.
16. Given:
Find:
17. Given:
Find:
mLRST=39°
mLTSV=23°
mLRSV
mLRSV=59°
mLTSV= 17°
mLRST
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CHAPTER 1 tJ LINE AND ANGLE RELATIONSHIPS
38
18. Given:
mLRST=2x+9
mLTSV=3x-2
mLRSV=67°
x
Find:
mLRST=2x-10
mLTSV=x+6
mLRSV = 4(x - 6)
x and mLRSV
19. Given:
Find:
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s~
Exercises 16-24
Find:
21. Given:
28. For two complementary angles, find an expression for the
measure of the second angle if the measure of the first is:
a) XO
b) (3x - 12t
c) (2x + 5yt
29. Suppose that two angles are supplementary. Find
expressions for the supplements, using the expressions
provided in Exercise 28, parts (a) to (c).
30. On the protractor shown,
mLRST= 5(x+ 1) - 3
mLTSV = 4(x - 2) + 3
mLRSV= 4(2x + 3) - 7
x and mLRSV
20. Given:
,,\
ii-iiiiiiiliiiiiiiiiiL;1l'ir:rl.. ;;;iiiiiiiiiimiiiiiiiiiiiiiiiii_iiiiiiiiiiiiiiii~~~~iiEliillliliI
NP bisects
LMNQ. Find x.
mLRST=~
\
mLTSV=~
mLRSV=45°
x and mLRST
Find:
Exercises 3D, 31
mLRST=~
22. Given:
"
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3
mLTSV=~
mLRSV=49°
x and mLTSV
Find:
31. On the protractor shown for Exercise 30, LMNP and
LPNQ are complementary. Find x.
32. Classify as true or false:
a) If points P and Q lie
lies in the interior of
b) If points P and Q lie
lies in the interior of
c) If points P and Q lie
lies in the interior of
--->
23. Given:
ST bisects LRSV
mLRST=x+y
mLTSV=2x-2y
mLRSV=64°
x andy
Find:
--->
24. Given:
ST bisects LRSV
mLRST = 2x + 3y
m L TSV = 3x - y + 2
mLRSV = 80°
x andy
Find:
<---->
25. Given:
+----->
in the interior of LABC, then PQ
LABC.
in the interior of LABC, then
LABC.
in the interior of LABC, then PQ
LABC.
PQ
In Exercises 33 to 40, use only a compass and a straightedge
to perform the indicated constructions.
<---->
AB and AC in plane P as shown; AD intersects
Pat point A
LCAB
LDAC
LDAC=
LDAB
What can you conclude?
=
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/
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~;
L-7~--
26. Two angles are complementary. One angle is 12° larger
than the other. Using two variables x and y, find the size
of each angle by solving a system of equations.
•
27. Two angles are supplementary. One angle is 24° more
than twice the other. Using two variables x and y, find the
measure of each angle.
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Exercises 33-35
33. Given:
Construct:
Obtuse LMRP
With
as one side, an angle
34. Given:
Construct:
Obtuse LMRP
RS, the angle bisector of LMRP
35. Given:
Construct:
Obtuse LMRP
Rays !?S, Iff, and
RfJ so
divided into four
= angles
36. Given:
. Construct:
oA
= LMRP
that LMRP is
Straight LDEF
A right angle with vertex at E
(HINT: Use Construction 4.)
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Introduction
1.5
37. Draw a triangle with three acute angles. Construct angle
bisectors for each of the three angles. On the basis of the
appearance of your construction, what seems to be true?
38. Given:
Construct:
:"42.
IfmLTSV=
38°, mLUSW=40°,
find mLUSV.
Acute L 1 and AB
Triangle ABC with LA "" L I, LB "" L 1,
and side AB
L
~-----.
B
A
39. What seems to be true of two of the sides in the triangle
you constructed in Exercise 38?
ED
"40. Given:
Straight LABC and
Construct:
Bisectors of LABD and LDBC
What type of angle is formed by the bisectors of the two
angles?
41. 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).
39
to Geometric Proof
and mLTSW=
61°,
u
v
IN
Exercises 42, 43
= x + 2z, mLUSV = x - z, and
43. IfmLTSU
mL VSW = 2x - z. find x if mLTSW = 60,
Also, find z if mL USW = 3x - 6.
44. 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).
!f-~+-----"J
v
T
8
A I?------<>---~
, 45. On the hanging sign, the three angles (LABD, LABC ,
and LDBC) at vertex B have the sum of measures 360°, If
mLDBC = 90° and SA bisects the indicated reflex angle,
find mLABC.
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Proof
Algebraic Properties
Given Problem and Prove
Statement
Sample Proofs
To believe certain geometric principles, it is necessary to have proof. This section introduces some guidelines for proving geometric properties. Several examples are offered to help you develop your own proofs. In the beginning, the form of proof will be
a two-column proof, with statements in the left column and reasons in the right column.
But where do the statements and reasons come from?
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