Download Geometry - Chapter 2 Day #1 Proofs

1
I
Geometry - Chapter 2 Day #1 Proofs
I.
Definitions
I
1.
Reasoning - process in which conclusions
Deductive
are drawn logically from
given information.
2.
II.
Geometry Proof - A valid argument that shows a mathematical
true: PROOF is the result of CORRECT deductive reasoning.
PROPERTIESFOR REAL NUMBERS:
statement
to be
1. Reflexive - says that an item is equal to itself.
2.
Symmetric
- says that we can switch the left side of equation
to the right side.
3. Addition - says we can add the same item from BOTH sides of an equation.
4. Subtraction - says we can subtract the same item from BOTH sides of an
5.
equation.
Multiplication
- says we can multiply
both sides of an equation
by the same
item.
6.
7.
Division - says we can divide both sides of equation
8.
Transitive - says we can throw out any item that appears in 2 separate
equations. (Use ONLY if no +, -, multiplication
or division is involved).
9.
Substitution
Distributive
- says we remove ( ) by multiplying
by the same item.
all items inside by the item
outside the ( ).
equations
- says we can throw out any item that appears in 2 separate
and replace it. (USE ONLY IF THERE IS AN OPERATION: +, -, multiply,
or divide).
10. Commutative property11. Associative broperty12. Adding/Combining
III.
Examples:
Like Terms -
State the reason for each statement below:
1. If 3x :::6, then x
= 2.
2. If <A + 3<A :::60, then 4 <A :::60.
3. If 3(x + 4}::: <B, then 3x +12::: <B.
4. <B::: <B
5. If 3x - 2 :::4, then 3x :::6.
6. If 4x + 1 :::5, then 4x :::4.
7. If CE = x and x = EF, then CE = EF.
1
8. If CE = x + 2 and = w, then CE = w + 2.
9. If <A = Y + 2 and kc = y + 2, then <A = <c.
10. If <A = Y +
C
and y = a + b, then <A::: a + b + c
11. If <C :::4w and 1B :::w, then <C :::4<B.
12. If <X::: <1 and <ll = <2, then <X = <2.
13. If x + 3 :::y, then y = x + 3.
14. If 7x = 28, then ~ = 4.
15. If RS = TU and TU = YP, then RS::: YP.
16. If VR + TV = EN + TV, then VR ::: EN.
t
17.lfa:::bandx=a,thenb=x.
18. If x + y :::5 and
= 6, then 6 + Y :::5
19. If x + w ::: 10 ana x = 8, then w = 2.
I
20. If BD :::x + 14 arid CE :::x + 14, then BD ::: CEo
2
COMPLETE ON A !SEPARATE SHEET OF PAPER:
HOMEWOR~ - Chapter 2 Day #1
Answer the following on the same sheet of paper as the problems from the textbook
A.
State the reafon for each statement
1.
2.
3.
4.
5.
6.
7.
8.
9.
If 2x = 5, then x = 5/2
If x/ 2 = 7, then x = 14.
If x = 5 and b = 5, then x = b.
If XV- AB 1= WZ - AB, then XV = WZ.
If m<A = m<B and m<B = m<C, then m<A = m<c.
If HJ + 5 =120,then HJ = 15.
If XV + 20 = VW and XV + 20 = DT, then VW = DT.
If m<l + m<2 = 90 and m<2 = m<3, then m<l + m<3
I
If Yz AB = ~ EF,then AB = EF.
= 90.
10. AB = AB I
11. If 2(x - 3/2) = 5, then 2x - 3 = 5.
12. If EF = JK alnd EF = GH, then JK = GH.
13. If 2x - 1 = 4, then 2x = 5.
14. <C = <C
15. If 4(x -1) J <C, then 4x - 4 = <C
16.
17.
18.
19.
20.
21.
If <B + <B ~ 50, then 2«B) = 50.
If 2«B) = 50, then <B = 25.
If 3x + 2 = 4, then 3x = 2.
If AB = 4x Jnd x = FG, then AB = 4 FG.
If XV = x + 3 and x = a, then XV = a + 3.
If <C = Yz a bnd a = b, then <C = Yz b.
22. If <W = Y +12and <W = Y + 3, then y +2 = Y + 3.
23. If XV = x and x = a, then XV = a.
24. If Y = 7 + x, ~hen 7 + x = y.
B.
Which property asserts that:
1. A number it equal to itself?
2. The sides of an inequality can be interchanged?
3. If a first nurber equals a second and the second number equals a third, then
the first number equals the third number.
c.
4. Vou can ad~ the same number to each side of an equation and the equality if
preserved?1
Name the pra4erty of equality (reflexive, symmetric, or transitive) that allows
you to conclude that each statement
1. If JK
= LM, t~en
LM
is true.
= JK.
2. If m~R = m<!ls,then <S = <R.
3. MN - MN.
4.
If AB = CD ard CD = EF,then AB
5. If x =~' thel113 = x.
6. m<C - m<c.
=
EF.
-1 Chapter
Geometry
1
2 Day #2 Proofs
The following theorems, postulates, and concepts MUST be memorized in
order to be able
understand and work proofs:
io
1. Definition
2. Midpoint
into 2
of Mid Joint - A point that divides a
Theore~
- If B is a midpoint
Definitio~
of comblementary
4. Definition
of Supplementary
of segment AC, then
angles -Two
= parts.
= Y2
angles that add up to equal 90 degrees.
angles - Two angles that add up to equal 180 degrees.
5. Definition
of Right!Angle - An angle that adds up to equal __
6. Definition
of Right Triangle - A triangle
7. Definition
of AnglJ Bisector - A ray that divides an angle into 2
that has __
degrees.
right angle.
parts.
= Y2
8. Angle Bisector The.orem - If ray BX is an angle bisector of <ABC, then
Definition
~f LineJ
10. Definition
11.
angles that form a
Isosceles Triangle! Theorem - If a triangle
Definition
sides.
is isosceles, then the base angles are
of Acute Angle - An angle that measures
than 90 degrees.
13. Definition
of ObtJse Angle - An angle that measures
14. Definition
of Perp~ndicular
15.
_
with at least two
I
congruent.
12.
Pair - Two
of Isosd:eles Triangle - A triangle
Linear Pair postul1te
-If
than 90 degrees.
Lines - Two lines that form
right angles.
two angles form a linear pair, then the two angles are
I.
16. Angle Addition
Po~tulate --
=
+
(in terms of angle
measures).
segment
measures).
18. Definition
=
+
17. Segment Additionl Postulate --
I
of Vertical Angles - Non-Adjacent
(in terms
angles formed
by two intersecting
I
19. Vertical Angle Theorem - If two angles are vertical angles, then the two angles are
______
to one another.
20. Substitution
21. Transitive
- uselthiS property
if an item is eliminated
- use if an item is eliminated
within
a math problem.
where no +, -, Multiplication,
or Division
occurs.
22. Adding Like Terms - Use to combine items on
23. Addition Property
Use to add an item to
side of an equation.
r
24.
25.
26.
27.
Subtraction
Properjty - Use to subtract
Division Property
Multiplication
Distributive
J Use to
proJerty
sides of an equation.
an item from
divide
- Use to multiply
sides of an equation.
sides of an equation
by a number.
sides of an equation
Properltv - Use to remove
.
28. Ruler Postulate - Fbr every point on a number line, there corresponds
number
coordinate.
29. Protractor
I
Postulate
= -- All
angles measure between
by a number.
__
and __
a
---
degrees.
Make sure you undbrstand the difference between a CONDITIONALvs. a
CONVERSE!
I
2
Write a reason for each statement below:
<1.
<1 and ~2 are vertical angles.
2. If <1 and <2 are vertical angles, then <1 == <2.
*
3. AX+XB ~AB.
I
\4. <3 <i
and
•
4
-f3
I
'J..
3;:
.!!!
are a linear pair.
>->
5. If <3 & <j are a linear pair, then <3 & <4 are supplementary.
6. If <3 & <4 are supplementary,
<7.
If
X is a ldPoint
then m<3 + m<4
= 180.
of CD, then CX '" XD.
Ot
8. X is a mid oint of CD.
<
9. IfXY
= YM, then
C
12. If .1ABCis la right triangle, then <8 is a right angle.
I
o
'f 13. If BD ...LBe, then <B is a right angle.
A-
I
~ 14. If <DYXis a right angle, then
" (v1
•
-CO
t?I
VB ...LXY.
15. If X is a mifpoint of AB, then AX
= Y\ AB.
16. If AX == XB, then X is a midpoint of AB.
<:
0
:~c
11. LlABC is a light triangle.
<
A
18. If <C and <D are supplementary,
X
then <C + <D = 180.
If <A + <B =; 180, then <A and <B are supplementary.
p
__._
.W
r:
~
tit
-
B
17. If <C and <f are a linear pair, then <C and <D are supplementary.
'19.
•
II
'X..,L
I
____
•
X
LlXYM is an isosceles triangle.
10 . .1XYMis a~ isosceles triangle.
<
H
3
~o. If <1 + <2 = 90,
\
21. If <1 an
then <1 and <2 are complementary.
<2 are complementary,
then <1 + <2
~ 22. <2 + <5 ~ <X~:~..=,.
----~~-
I
23. <A is an acute
angle.
24. If <A =
?!
~
9b. then <A is a right angle.
25. If <A is a Iright angle, then <A = 90.
lr
26. If <A = 45 degrees, then <A is an acute angle.
27.If<B
=
I
.
degrees. then <B IS an obtuse angle.
28. <B is an obtuse angle.
II. Proofs:
Solve the followi1g Algebra proofs:
1. If 3(n + 1] = 99, then n = 32.
2. If 2x + y
= ~3, and
3. If -1/2 x
= Ll,
y
then x
= 7, then
= -22
x
= 8.
= 90.
1111
I
4
Day #2 Proofs HOMEWORK
I.
S
Write a ~eason for each sta~ement belo~ ~
lj 1. ~ABCI is an isosceles triangl~_
A-
<2
"-I
.. " .
M is the midpoint of AB.
I
C
A __
-
3.
If M is a midpoint of CD, then CM
4.
<1. + 42 = <ABC.
== MD
H
tn
"
13
A
I
P
:f7
~
r.....
#£"
5.
,1DEF is a right triangle
6.
<1 and <2 are v~t~angles.
7.
~
<3 and <4 are a linear pair.
• . .,.,.,
3A
e
;p
8.
If <2 +1<3 = 180, then <2 and <3 are supplementary.
9.
If <3 and <5 are supplementary,
then <3 + <5 = 180.
10. If <B =i101, then <B is an obtuse angle.
rr
c
A-To
11. If <1 arid <2 are vertical angles, then <1 ~<2.
12. IfCEJjEB,then
<E is a right angle.
13. If <E = 90, then <E is a right angle.
14. If <E is
a right angle, then
<E = 90.
15. If <C and <E are complementary,
then <C + <E = 90.
16. If <3 + 14 = 90. then <3 and <4 are complementary.
17. If <2 an6 <3 are a linear pair, then <2 and <3 are supplementary.
1
18. If <2 = 'X8, then <2 is an acute angle.
19. AC + CD'= AD..
I
20. <ABC = rABc.
II.
•
ACt>
Algebra Proofs.
1.
2.
Ifx + y =13 and y = -2, then x = 5.
If 4(x - 1) = 90, then x = 23.5.
I
3. If -3x + 1 = 10, then x = -3.
_
.,.13
1
Geometry - C~apter 2 Extra Practice on Properties
State the reason for each Df the following:
= 6, then x 1-3.
1.
If x + 9
2.
AB + BC = AC (in thb figure shown):
~
~
•
A- B
c
=
3.
4.
5.
6.
<ABC <ABC
Ifx = 6 and x = V, t~en V = 6.
Ifx=6andw+x=121,thenw+6=21
<1 + <2 = <WXV (in Ithe figure shown):
7.
If <1 is a right angle, then
I
~
X
~
AB..L XY (in the figure shown).
I'
A •
8.
If x + a = 41 and x +
9.
If w + Z = 17 and
Z
r, = z, then
Z
.• g
'(
= 41.
=19,then w = 8.
10. If 9(x + 6) = 21, the19X + 54 = 21.
--.
I
11. If B is the midpoint of Al, then AR
.- = R7
-12. If B is the midpoint ~f Al, then AB =
7S Al.j
••
Z
•
A
B
13. If}\ AZ = AB, then B Is a midpoint of AZ. ,;
14. If <1 and <2 are a Itar
I
pair, then <1 and <2 are supplementary.
15. If <1 + <7 = 90, then 1"1 & <7 are complementary.
r
17. <M is a right angle.
18. If <M = 80, then <M 1san acute angle
19. If ~ + 7~Z
and V, = 15, then MZ = 22.
20. If XY
..L
~
,
AB, then <3 IS a right angle (in the figure shown).
21. <A + 6<A + 10<A =
n <A.
A
,
:3
B
x
I
e
=
22. If BO is an angle bilector, then <1 <2. (in the figure shown)
23. If <1 = <2, then BO is an angle bisector.
24. If BO is an angle bi. ector, then <1 Yz <ABC.
2.o".
~
=
25. If <2
= Yz <ABC, the'n
BO is an angle bisector.
~
26. <1 and <2 are verti~al angles (in the figure shown) ~
27. If <1 and <2 are vetical
28. B is a midpoint
angles, then they are congruent to one another.
(in the figure shown)
I
29. BC + CO = BO (in thi figure shown)
=
30. If 5x 25, then x = 5.
31. If 2x + 10 ::;26, the~ x
=
••.•
Q
.13-
J3
c•
= 8.
=
=
32. If <1 20 and <2 20, then <1 <2.
33. If <1 is a right angle] then QR...L PV (in the figure shown)
34. If <6 + <9
i-.
~
p
Q •.
D
*
I,
v
= 90, thenl <6 and <9 are complementary.
35. If <1 and <8 are veiieal
36. If x/2
H
It·
angles, then <1 = <~
= 9, then x = 18.
I
+ 6 = 20, then x = 28.
37. If x/2
38. If <1 & <10 are a linJar pair, then <1 and <10 are supplementary.
39. If A + B = Wand B = C, then A + C = W.
,j
40. If <6 = <18 and <181 <7, then <6
41. If 4(x + z) ::;5h, then 4-x+ 4z = Sh.
= <7.
42. <5 + <7 = <KlZ (in the figure shown):
43. JK + KM
44. <5 &
45. If AB
= JM
?to
L-£
V\
--
-,
(in the figure shown):
are a linearlpair (i~he
~<'HIt, then
figu!:'shOWn):
<81 is a right angle, (in the figure shown):
A
K
p,
3
~
~
46. If <1 = 156, then 11 is an obtuse angle.
47. <5 is an acute angie.
48. If WK = Yz WZ, theh K is a midpoint (in the figure shown):
- K-
fI
w
49. If <1 = <2, then HK,is an angle bisector (in the figure shown):
~
50. If AB + TU = AB + WZ, then TU = WZ
51. If <5 + <7 = 180, thb n <5 + <7 are supplementary.
52. If 5X + 6X + 100X = 10445, then l11X = 10445.
l
~:::;:;: :::~~::c:t~~entary,then<8 <14 90
+
=
56. If <B = 90, then <B is a right angle.
57.
58.
59.
60.
If <1 & <15 form a linear pair, then <1 & <15 are supplementary.
If <1 & <15 are sup~'ementary, then <1 + <15
If <6 & <7 are verti2al, then <6 = <7.
<7 + <9 = <YKZ(in t~e figure shown). -
Answers:
#1. Subtraction
#3. Reflexive property
#5. Substitution
#7. Converse of Definition of perp~ndicular
#9. Substitution & subtraction
#11. Definition of midpoint
#13. Converse of midpoint theorem
#15. Converse of Definition of Com~lementary
#17. Definition of Right angle
#19.CLT & Substitution
#21. CLT
#23. Converse of Definition of Angle Bisector
#25. Converse of Angle Bisector ThJorem
#27. Vertical Angle Theorem
#29. Segment Addition Postulate
#31. Subtraction & division
#33. Converse of Definition of Perpendicular
#35. Vertical Angle Theorem
#37. Subtraction & multiplication
#39. Substitution
#41. Distributive
#43. Segment addition postulate
#45. Definition of perpendicular
#47. Definition of acute angle
#49. Converse of definition of angle bisector
#51. Converse of definition of suppl~mentary
#53. Subtraction
#55. Definition of complementary
#57. Linear pair postulate
#59. Vertical angle theorem
I
= 180.
-Y'
~
K
#2. Segment addition postulate
#4. Transitive
#6. Angle addition postulate
#8. Transitive
#10. Distributive
#12. Midpoint theorem
#14. Unear pair postulate
#16. Angle addition postulate
#18. Converse of Definition of Acute Angle
#20. Definition of Perpendicular
#22. Definition of Angle Bisector
#24. Angle Bisector Theorem
#26. Definition of Vertical Angles
#28. Definition of Midpoint
#30. Division
#32. Transitive
#34. Converse of Definition of Complementary
#36. Multiplication
#38. Unear pair postulate
#40. Transitive
#42. Angle addition postulate
#44. Definition of linear pair
#46. Converse of definition of obtuse-angle
#48. Converse of midpoint theorem
#50. Subtraction
#52. CLT
#54. Addition
#56~ Converse of definition of right angle
#58. Definition of supplementary
#60. Angle addition postulate
2..
1
Geometry -IChapter 2 Day #3 Proofs
Today, we will spendj 0 Jot of time reviewing angle relationships that will playa big role
in the proofs that we are about to encounter.
I.
Short Answer
Name two pairs of congruent
#1.
angles in each figure.
Justify your answers.
#2.
B
#3.
K
: ../,/'
c
D
P
L
~t?
M
II. Problems
For each problem:
a. Identify the angle relationship that exists
b. Solve the problem for the given variable AND/OR given angles.
c. Identify the postulate, theorem, or definition that was used to solve the
problem.
#2. If m<2
#3. m<2
= 57, find\
#5. m<l
= 38, m<2
= 20, find
m<l.
<1
#6. m<13
find
#7. If m<9 + m<10;= 90. <7
= <9; <8 = 41, find
= 4x + 11, m<14 = 3x + 11,
X,
m<13 & m<14.
<9 & <10
#8. m<2 = 4x - 26, ~<3 = 3x + 4, find x and m<2 & m<3.
2
#9. m<5 = 5x, m16 = 4x + 6, m<7 = lOx, m<8 = 12x - 12,
find x and
mis,
m<7, & m<8.
III. Systems of Equations
In these problems, a system of equations may be needed to find the variables and
the angle measure.
#1.
#2.
2x
I
O
4y~(X
IV. Word Problems
Find the measure
hi each angle.
1. <A is twice as large as its complement,
2. <A is half as large as its complement,
<B.
<B.
3. <A is twice as large as its supplement, <B.
4. <A is half as large as twice its supplement, <B.
+Y +
10)0
3
I
~O~h~~?~J~-
Chapter 2 Day #3 Proofs
Name two pairs of cbngruent angles in each figure. Justify your answers.
#1.
#2.
e
W
y---=Q
E
D
c
x
II.
A
Problems:
For each problem:
a. Identify the angle relationship that exists
b. Solve the problem for the given variable AND/OR given angles.
c. Identify the postulate, theorem, or definition that was used to solve the
problem.
#1. If <2 = 67, find m<1.
#3. If m<7 + m<8
= 90,
#2. If m<3 = 38, find m<4.
m<5
= m<8, and m <6 = 29, find
#4. If m<9 = 4x - 4 and m<10 = 2x + 4, find
= 4x and m<12 = 2x -
#5. If m<l1
6, find
X,
X,
m<9, and m<10.
m<l1, and m<12.
Systems of Equations.
III.
#1.
(x
+y +
5)°
yO X 2xO
m<5, m<7, & m<8
#2.
4
IV.
Word Problems
Find the measure of fach angle.
1.
2.
3.
4.
V.
<A
<A
<A
<A
is three times as large as its complement, <B.
is 21 less than twice as large as its supplement, <B.
is congruent to its supplement, <B.
is 18 more than five times its supplement, <B.
Multiple
Choice
Choose the correct letter.
1. The measure of <B is one-half the measure of its complement.
measure of <B.
What is the
a. 30
b. 45
c. 60
d. 90
2. <T and <R are vertical angles. m<T:;; 3x + 36 and m<R
measure of <I?
a. 15
b. 81
c. 87
d. 99
= 6x -
9. What is the
i
(6x - 5)°
Use the figure to answer the next TWO questions:
(2y + 3~3Y
3. What is the value of x?
(8x -
a. 8.9
+ 20)0
5W
b. 16.8
c. 22.5
d. 27.5
4. What is the value of y?
a. -10
b. 2
c. -2
d. 10
5.
<A and <B are complementary
the value of x?
a. 3
b. 6
c.
11
d. 22.25
I
angles. If m<A
= 5x -
2, and m<B
= 3x + 4, what
is
1
Geometry - C~apter 2 Day #4
Topic: Conditional Statements & Biconditionals
Today, we are going to mdve further into the world of conditional statements from where we started in
the previous chapter. We learn these things to help increase our understanding of 'reasoning skills'.
Definitions:
1. Conditionala.
b.
2.
3.
4.
an 'if-then'
Symbols: p ~ q
This reads as: "if
statement
'pi then 'q'" OR as IIIp' implies 'q'".
Hypothesis - is the part 'p' following if
Conclusion - is the part 'q' following then.
Truth value - is whether the conditional is true or false.
EXAMPLES:
#1. What are the hypothesis and the conclusion of the conditional?
"If an animal is a robin, then the animal is a bird."
"If an angle measures 130, then the angle is obtuse."
#2. How can you write the following statement
"Vertical angles share a vertex."
as a conditional?
"Dolphins are mammals"
#3. Is the conditional true or false? If it is false, find a counterexample.
"If a woman is Hungarian, then she is European."
"If a number is divisible by 3, then it is odd."
"If a month has 28 days, then it is February."
"If two angles form a linear pair, then they are supplementary."
We also want to make sure we know the difference
in several different
related conditional
statements.
Negation - The negation of a statement 'p' is the opposite of the statement. The symbol we use to
illustrate this is ~p and is read 'not p'. Ex: The negation of 'the sky is blue' is 'the sky is NOT blue'
STATEMENT
How to Write it
EXAMPLE
SYMBOLS
How to
Read It
Truth
Value
Conditional
Use the given hypothesis and
conclusion
If m<A = 15, then
<A is acute.
p~q
TRUE
Converse
EXCHANGE~he hypothesis and
the conclusion.
NEGATE both the hypothesis
If <A is acute,
then m<A = 15.
If m<A 1"- 15, then
q~p
If p, then
q.
Ifq, then
p.
If not p,
FALSE
and the con1clusion of the
conditional
<A is not acute.
NEGATE both the hypothesis
and the conbiusion of the
If <A is not acute,
then m<A 1"- 15.
Inverse
Contra positive
converse
~p~~q
FALSE
then not
q.
~q~~p
If not q,
then not
p.
TRUE
2
EXAMPLES: What are the Lnverse, inverse, and contrapositive of the following
the truth values of each? If a statement is false, give a counterexample.
"If the figure is a square, then the figure is a quadrilateral."
conditional?
What are
I
"If a vegetable is a carrot, then it contains beta carotene."
We also want to be able to write biconditionals
and recognize good definitions.
Biconditionalis a single true statement that combines a true conditional and its true converse. You
can write a biconditional by joining the two parts of each conditional with the phrase 'if and only if.'
A biconditional combines p ~ q and q ~ pas p ~ q.
EXAMPLE: 'A point is a midpoint if and only if it divides a segment into two congruent segments.'
EXAMPLES: What is the converse of the following true conditional?
the statements as a biconditional.
If the converse is also true, rewrite
#1. If the sum of the measures of two angles is 180, then the two angles are supplementary."
#2. "If two angles have equal measure, then the angels are congruent."
EXAMPLES: What are the two conditional statements that form this biconditional?
#1. "A ray is an angle bisector if and only if it divides an angle into two congruent angles.
I
#2. "Two numbers are reciprrcalS if and only if their product is 1."
3
EXAMPLES: Is this definition of quadrilateral reversible?
#1. "A quadrilateral is a p61ygon with four sides."
If yes, write it as a true biconditional.
#2. "A straight angle is an angle that measures 180 degrees."
#3. "A Tarheel is a person who was born in North Carolina."
We also want to be able to determine when a statement is not a good 'definition.'
is by finding a counterexample.
MULTIPLE CHOICE EXAMPLE: Which of the following
is a good definition?
a. A fish is an animal that swims
b. Rectangle have four corners
c. Giraffes are animals with very long necks
d. A penny is a coin wqrth one cent.
EXAMPLE: Is each statement a good definition?
1. A square is a figure with four right angles.
2.
A segment is a part of a Ine.
3.
Opposite rays are t'40 rays that share the same endpoint.
4.
A cat is an animal with whiskers.
5.
The red world if an erdangered
6. A compass is a geometric tool.
animal.
One way to do this
4
HOMEWORK
+ Chapter 2 Day #4
Identify the hypothesis and conclusion of each conditional.
1. If a number is a multiple of 2, then the number is even.
2. If something is thrown up into the air, then it must come back down.
3. Two angles are supplementary
if the angles form a linear pair.
4. If the shoe fits, then you should wear it.
State whether each conditional is true or false. Write the converse for the conditional and state
whether the converse is true or false.
5. If the recipe uses 3 teaspoons of sugar, then it uses 1 tablespoon of sugar.
6. If the milk has passed its expiration date, then the milk should not be consumed.
State whether each conditional is true or false. Write the inverse for the conditional and state
whether the inverse is true or false.
7. If the animal is a fish, then it lives in water.
8. If your car tires are not properly inflated, then you will get lower gas mileage.
State whether each conditional is true or false. Write the contrapositive for the conditional and state
whether the contra positive is true or false.
9. If you ride on a roller coaster, then you will experience sudden drops.
10. If you only have $15, then you can buy a meal that costs $15.65.
I
Test each statement below to see if it is reversible. If it is reversible, write it as a true biconditional. If
not, write not reversible.
11. If a whole number is a multiple of 2, then the whole number is even.
12. Rabbits are animals that eat carrots.
13. Two lines that intersect to form four 90° angles are perpendicular.
14. Mammals are warm-blooded
animals.
Write the two conditionals that form each biconditional.
15. parallelogram
16.
is a rectangle if and only if the diagonals are congruent.
An animal is a giraffe if and only if the animal's scientific name is Giraffe camelopardalis.
State whether each statement is a good definition. Explainyour answer.
17. A parallelogram is a quadrilateral with two pairs of parallel sides.
18.
19.
20.
21.
A triangle is a three-sided figure whose angle measures sum to 180°.
A juice drink is a beverage that contains less than 100% juice.
In basketball, the top scorer in a game is the player who scores the most points in the game.
A tree is a large, green, leafy plant.
Geometry .L Chapter 2 Day #5
Topic: Deductive Reasoning
Today, we are going Ito learn some more things about deductive reasoning.
Specifically, we will/earn to use the Law of Detachment and the Law of Syllogism.
Law of Detachment states that if a conditional statement is true, then any time the
conditions for the hybothesis exist, the conclusion is true.
If the conditional statement is not true, or the conditions of the hypothesis do not
exist, then you cannot make a valid conclusion.
Problem
What can you conclude from the following series of statements?
~ If an animal has feathers and can fly, then it is a bird.
~ A crow has feathers and can fly.
o Is the conditional statement true? Yes.
o Do the conditions of the hypothesis exist? Yes.
o Therefore, you can conclude that a crow is a bird.
~ If an animal has feathers and can fly, then it is a bird. A bat can fly.
o Is the conditional statement true? Yes.
o Do the conditions of the hypothesis exist? No; a bat does not have
feathers.
o Therefore, no conclusions can be made with the given information.
EXAMPLES:
Use the Law of Detachment to make a valid conclusion based on each conditional.
Assume the conditional statement is true.
1.
If it is Monday, then Jim has tae kwon do practice.
The date is Monday, August 25.
2.
If the animal is a whale, then the animal lives in the ocean.
Daphne sees a beluga whale.
3.
If you live in the city of Miami, then you live in the state of Florida.
Jani lives in Florida.
4.
If a triangle has an angle with a measure greater than 90, then the triangle is
obtuse.
In 8GHI, mLHGI 110.
=
5.
A parallelogram is a rectangle if its diagonals are congruent.
lincoln draws a parallelogram on a piece of paper.
p-+q}
ci:
r
I
TherfOre: p -+ s
~S
~
~
You can use the Law of Syllogism to string together two or more conditionals and draw a
conclusion based on the conditionals.
Notice that the conclusion of each conditional becomes the hypothesis of the next
conditional.
Problem
.:.
What can you conclude from the following two conditionals?
o If a polygon is a hexagon, then the sum of its angle measures is 720.
o If a polygon's angle measures sum to 720, then the polygon has six
sides.
• The conclusion of the first conditional matches the
hypothesis of the second conditional: the sum of the angle
measures of a polygon is 720.
• You can conclude that if a polygon is a hexagon, then it has
six sides.
EXAMPLES:
If possible, use the Law of Syllogism to make a conclusion. If it is not possible to
make a conclusion, tell why.
6.
If you are climbing Pikes Peak, then you are in Colorado. If you are in
Colorado, then you are in the United States.
7. If the leaves are falling from the trees, then it is fall.
If it is September 3D, then it is fall.
8. If it is spring, then the leaves are coming back on the trees.
If it is April 14, then it is spring.
9. If plogs plunder, then flegs fret.
If flegs fret, then gops groan.
10. If you make a 95 on [the next test, you will pass the course.
If you pass the course, you will not have to take summer school.
11. Use the Law of DetaFhment and the Law of Syllogism to make a valid conclusion
from the following series of true statements. Explain your reasoning and which
statements you used in rour reasoning.
I. If an animal is an insedt, then it has a head, thorax, and abdomen.
II. If an animal is a spider, then it has a head and abdomen.
III. If an animal has a head, thorax, and abdomen, then it has six legs.
IV. Humberto saw an insect called a grasshopper.
HOMEWORIK - Chapter 2 Day #5
If possible, use the Law of Detachment to make a conclusion. 1/it is not
possible to make a conclusion, tell why.
1. If a triangle is a ri~ht triangle, then the triangle has one 90Q angle. MBC is a
right triangle.
2. If a parallelogram
has four congruent
sides, then the parallelogram
is a
rhombus.
The parallelogram
3.lfx>7,then
has four congruent
sides.
[x] >7.
x<7
4. If cats prowl, mice will scatter.
Mice are scattering.
5. If the light is flashing yellow, then you may drive with caution through the
intersection.
The light is flashing yellow.
6. If a triangle has two congruent
-
-
sides, then the triangle is isosceles. In j}"DEF,
DE::::EF.
1/possible, use the Law of Syllogism to make a conclusion. 1/it is not possible
to make a conclusion, tell why.
7. To take Calculus, you must first take Algebra 2.
To take Algebra 2, you must first take Algebra 1.
8. If a tree has ragged bark, then the tree is unhealthy.
If a tree has ragged bark, then the tree might be a birch tree.
9. A quadrilateral has four congruent
A square is a rhombus.
sides if and only if it is a rhombus.
Using the statements below, apply the Law of Detachment or the Law of
Syllogism to draw a conclusion.
10. If Jorge can't raise money, he can't buy a new car. Jorge can't raise money.
11. If Shauna is early for her meeting, she will gain a promotion.
If Shauna
wakes up early, she will be early for her meeting. Shauna wakes up early.
12. If Linda's band wins the contest, they will win $500. If Linda practices, her
band will win the contest. Linda practices.
13. If Brendan learns the audition
song, he will be selected for the chorus. If
Brendan stays after school to practice, he will learn the audition
Brendan stays after school to practice.
song.
ForExercises 14-17, apply the Law of Detachment, the Law of Syllogism, or
both to draw a conclusion. Tell which Jaw(s) you used.
14. If you enjoy all foods, then you like cheese sandwiches.
If you like cheese
sandwiches, then ybu eat bread.
15. If you go to a monster
movie, then you will have a nightmare.
You go to a
monster movie.
16. If Catherine is exceeding the speed limit, then she will get a speeding
ticket. Catherine is driving at 80 mi/h. If Catherine is driving at 80 mi/h, then
she is exceeding the speed limit.
17. If Carlos has more than $250, then he can afford the video game he wants.
If Carlos worked more than 20 hours last week, then he has more than $250.
If Carlos works 15 hours this week, then he worked more than 20 hours last
week.
Chapter 2 Day #6 Proofs Examples
Homework: WorksHeet Proofs #1-#4
Topic: Algebra + Gebmetry Proofs
p
1. Given: C is between Band D
BC= 8x
CD = 2x- 3
BD= 34
Prove: BC= 29.6
e
c
(L
2. Given: Q is the midpoint of PR
PQ= 4x-l
QR = 9x-14
Prove: PR= 18.8
3. Given: m<XYW = 90
m<l = 4x + 5
m<2 = x- 3
Prove: m<l = 75.4°
t'
x
'(
4. Given: <PQR is a right angle
m<1=7x~4
m<2 = 2x + 5
Prove: m<2 = 23°
Q
A/,
\IV
p
Q
p,
Chapter 2 Day #7 Proofs Examples
Homewor~: workshJet Proofs #5·#8
Topic: Segment + Midpoint Proofs
1. Given: PR= 88
I PQ= 4x + 3
QR = 8x-38
Prove: Q is the midpoint of PR
2. Given: BD = AC
Provb: CD = AB
f7
Q..
F
~~
~..,
A
C
S
!'>
3. Given: F is between A and B
FA= 24
~
{>t
FB = 4x + 8
AB = 12x
Prove: F is the midpoint of AB
.p
I
4. Given: B is the midpoint of AC
c
BD = AB
Prove: ~ AC = BID
I
A
nu
"1-
U
ct
a
1\
•
\\
I
I
I
Chapter 2 Day #8 Proofs Examples
Homework: Worksheet proofs #9-#12
Topic: Complementdrv Angles + Perpendicular Lines Proofs
1. Given: <X & <Yare complementary
<X = 3<Y
Prove: m<Y = 22.5°
'f..
--'>
--'>
2. Given: QP...L QR
m<l = 48°
Prove: m<2=42°
'f'
P
(Q
p,
3. Given: <1 & <2 are complementary angles
--'>
---->
Prove: AB...L AC
B
--'>
---->
4. Given: GB...L GC
A
Prove: <1 & <3 are complementary
A
b
e
c
I
I
Chapter 2 Day #9 Proofs Examples
E
Homework: Worksheet Proofs #13-#16
I
Topic: Linear Pairs + Vertical Angles Proofs
II
1. Given: m<EFA"" m<6
Prove: m<5"" m<l
v, -
<15-
..~-• ~
2. Given: m<2:::: m<8
Prove: <1 & <8 are supplementary angles
~VJl:7
1_...-..-------7
"
t:='
3. Given: <4:::: <5
Prove: <DeE:::: <4
..
A -
-
D
.•.
C
4. Given: <1:::: <2
Prove: m<2 + m<3 = 180
I
~
;;~
-_
~
.. -
'
~
~
~)I"t/.A~w")lddl\S
=\'" ~
I~};> ~ ~
I~
: -1J(\O.J.dj
f 14't1 e.T- : 'I}.~(\,I
<;-=If
~I
Ii
ChaPt,J 2 Da~ #10 Proofs Examples
Homework: WorksHeet Proofs #17-#20
•
III
•
I
TopIC: Angle BIsectors + Angles Measures Proofs
1. Givln:
I~
m<2:::
m<3
Prove: BP bisects <ABC
2. Given: <XYZ = <wyv
Pro~e: <7 = <5
y "IEZ= 7
I'
W
,..."
v
3. Givl:
QiJ bisects
---->
<NQP;
Ii
QN bisects <MQO
Prove: <5 = Yz <MQO
r
4. Given: <A & <B are supplementary
<A
Provt
angles
= 4<B + 7
<A is an obtuse angle
A
B
-
~.
\J
LU
ro
\J
..at
~\
~
~"
()
CA
<t:
~
.;.H
Jt :r
~
..
~ ~
?
-'t
- .•
~
~
._---
'-.00-
*{
*
.Q
~~
2
Due Date:
Chapter 2 Proofs - Homework
----
6. Given: XW == YZ
Prove: XY == WZ
Directions:
1. Complete all proofs using the 2-column format on a separate sheet of
paper.
2. Draw all the pictures and 'mark' them with given information.
7. Given: D is between A and C
3. Complete Proofs in chronological order.
4. You should complete 2
proofs per side of sheet of paper
AD =4x -15
(no more, no less).
CD
(i.e.: 1 sheet of paper should have 4 proofs on it; 2 on front; 2 on back)
Proofs:
1. Given: M is between A and C
---AM-=-4x
MC=2x-3
AC
,
•
4
= 33
Prove: AM = 24
= x + 15
AC= 50
Prove: D is the midpoint
••
c
8. Given: P is a midpoint
of AC
2. Given: X is a midpoint of AB
o
AX = 3x-1
XB
= 9x -40
Prove: AB = 37
A
---Prove:
J3
= 71'
4. Given: <DEF is a right angle
<3 = x + 10
<4 = 2x + 20
<3=30·
9. Given: <B & <C are complementary
m-cll = 5<C
Prove: m<C = 15
4
3. Given: m<ABC = 90'
m<1 = 3x + 5
m<2 = x- 3
Prove: m<1
10.
11\
E
Given:
M
.1
BC
m<1 = 68'
V
5. Given: E is between D and F
p
vv
s
C-
A
Prove: m<2 = 22'
•• -----
••• --~
11.
DF =44
----~EF
H
of OQ
MN==PQ
Prove: Y, OQ = MN
DE = 3x + 1
= 4x - 6
Prove: ciS1lrniap-oint---
p
E
F
B
Given: <3 & <4 are complementary
Prove: MA -L MN
angles
4
12.
13.
14.
Given: AW..L AB
Prove: <4 & <6 are complementary
angles
17.
Given: m<4 = m<6
Prove: WR bisects <QRS
18.
Given: <BED = <AEC
Prove: <1 = <3
19.
Given:
Given: m<l = m<4
Prove: m<2 = m<3
Given: m<3 = mcS
Prove: <3 & <8 are supplementary
..te.-_-
BY
bisects <XBZ;
bisects <YBA
Prove: <10 = 1;2 <XB~
BZ
15.
Given: m<3
Prove: m<3
= m<2
= m-cl
4
20.
Given: <X & <Yare supplementary
m<X = 3·m<Y
Prove: <Y is an acute angle
angles
16.
y
5
I~
I
Bonus #·11:
Gh~len:X is thf midpoint of AB
AX=13K-41
BX = 5xl+ 12
Prove: AB = 90.25
Bonus #~:
Givtn: m<2 =m<7
Prove: m<3 = m<6
0,
I
Bonus #31:
Givcln: <C & <IDare supplementary
m<C
--->
= ml<D
l-.
angles
A
F
Prove: CA 1. CB
I: ----+
-L
: AB ..LAD
Pro
I
<1 & <3 are complementary
: m<2 = m-c l
B
E
o
Bonus: #
Given: <B is a right angle
<C is a right angle
<1 = <4
Pro~t <2 = <3
~
G
E.
•
'P
Geometry - Chapter 2 Test Review
(You will complete 6 proofs at 20 points each).
~ There will be ONEproof representing the following:
a Vertical Angles
a Midpoints
a Segments (part + part = whole)
a Linear Pair
a Complementary Angles & Perpendicularity
a Angle Bisectors
At least one proof will incorporate "algebra" in it.
___
---'-l.'---'-'G'-"~VEN;
K is the midpoint
JI=JI,JL
PROVE:
of JL
JI = LK
St
1. K is the midpoint of JL;
R
1. Given
JI=JI,JL
2.
_
2.
_
3.
_
3.
_
----~2~GIVEN:
PROVE:
PR
= ST; 5 is the
PR
= RS
midpoint
of RT
Q~----.~
St
R
2.
_
2.
_
3.
_
3.
_
5. GIVEN: AC = BD
PROVE: AB = CD
••
•
J3
A
St
•c ••0
X-
3
fA
7. GIVEN: <8 & <7 are complementary
PROVE:
YX
-L
IT
'{
1. AC = BD
R
1. Given
1. <8 & <7 are complementary
R
1. Given
2.
2.
2.
2.
3.
3.
3.
3.
4.
4.
5.
5.
6
6.
St
6. GIVEN: AB=3x+15,AC=90,BC=5x-5
A\
PROVE: BC = 45
8.
GIVEN:
YX
-L
X
IT
PROVE: <8 & <7 are complementary
St
R
1. AB = 3x + 15, AC = 90, BC = 5x - 5
1. Given
2.
2.
_
3.
_____
_
3.
_
_
4.
_
4.
_
5.
_
5.
_
6.
_
6.
7.
7.
St
1.
_
_
XV
-L
"
R
IT
2.
2.
3.
3.
4.
4.
5.
5.
6
G
J
9.
GIVEN:
QY bisects
PROVE: <6
<XMZ
= <3
l1.GIVEN:
<3
= 3x + 14, <4 = 5x -
PROVE: <3
R
St
1.
QY bisects
St
<XMZ
1.
2.
<l. = 2x + 14, <4 = 5x -
38
R
1. Given
2.
3.
3.
4.
4.
5.
5.
QY bisects
10.GIVEN:
PROVE: <6
= Y, <XMZ
QY bisects
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
<XMZ
R
St
1.
38
=.t(lO
<XMZ
1. Given
2.
2.
3.
3.
4.
4.
5.
5.
12. GIVEN: <5
PROVE: <6
= <6
= <7
St
1. <5
= <6
2.
2.
3.
3.
4.
-4.--