Deductive Reasoning “The proof is in the pudding.” “Indubitably.” Je solve le crime. Pompt...
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Transcript of Deductive Reasoning “The proof is in the pudding.” “Indubitably.” Je solve le crime. Pompt...
Deductive Reasoning
“The proof is in the pudding.”
“Indubitably.”Je solve le crime. Pompt de pompt pompt."
Le pompt de pompt le solve de crime!"
2-4 Special Pairs of Angles WE
2-4 Written ExercisesDetermine the measures of the complements and supplement of each angle. measure sums up to 900.
1 20m K
Complementary Supplementary
90 – 20 = 70 180 – 10 = 170
2
4
3
90 – 72.5 = 17.5 180 – 72.5 = 107.5
90 – x 180 – x
90 – 2y 180 – 2y
172
2m K
m K x 2m K y
5
6
2 complementary angles are congruent.Find their measures.
x + x = 90
2x = 90 x = 45
450 and 450
2 supplementary angles are congruent.Find their measures.
x + x = 180 2x = 180
x = 90
900 and 900
Vertical Angle Th.
Z
Y
XW
VU
TS
In the diagram, bisects andOT��������������
SOU35m UOV
120m YOW
Label completely !
Vertical Angle Th.
35
Vertical Angle Th.
120
6060
6060
60+60+35+x = 180 x = 25
25
25
35
Vertical Angle Th. Now you answer the questions.
O
Z
Y
XW
VU
TS
In the diagram, bisects and
OT��������������
SOU35m UOV
120m YOW
3560
60
606025
25
35
m ZOY
14
13
m ZOW
35
155
O
Z
Y
XW
VU
TS
In the diagram, bisects and
OT��������������
SOU35m UOV
120m YOW
3560
60
606025
25
35
m VOW
16
15
m SOU
25
120
O
Z
Y
XW
VUT
S
In the diagram, bisects and
OT��������������
SOU35m UOV
120m YOW
3560
60
606025
25
35
m TOU
18
17
m ZOT
60
85
O
22
are supplements
are supplements
1 2and
3 4and
a] If ,
find .
2 1 4 3
1 3 27m and m 2 4m and m
27 27
180 – 27 = 153
22
are supplements
are supplements
1 2and
3 4and
b] If ,
find .
2 1 4 3
1 3m and m x 2 4m and m
x x
180 – x
22
are supplements
are supplements
1 2and
3 4and
c] If 2 angles are congruent, must their supplements be congruent?
2 1 4 3x x
YES !
y y
23 Given:
Prove: 1 4and 2 3and
Label completely first. 12 3
4
g g
? ?
Statements Reasons
1 2and 2 3and 3 4and 1 4and
Note the flow is better without the given first.
Transitive Prop. Of Equality
Vert. Angles are congruent
Given
Vert. Angles are congruent
24 If and are supplementary,Then find the values of x, and .
A Bm Bm A
2 , 15m A x m B x
Start with a labeled diagram.
A B2x x - 15
2x + x – 15 = 180
3x – 15 = 180
3x = 195
x = 65Divide by 3
A = 2(65)
A = 130
B = 65 - 15
B = 50
25 If and are supplementary,Then find the values of x, and .
A Bm Bm A
16, 2 16m A x m B x Start with a labeled diagram.
A BX + 16 2x - 16
X + 16 +2x– 16 = 180
x = 60
Divide by 3
A = 60 + 16
A = 76
B = 2(60) - 16
B = 120 - 16
3x = 180
B = 104
26 If and are complementary,Then find the values of y, and .
C Dm Dm C
3 5, 2m C y m B y Start with a labeled diagram.
CD
3y+5
2y
3y + 5 + 2y = 90
5y + 5 = 90
5y = 85
y = 17
divide by 5C = 3(17) + 5C = 51 + 5
C = 56
D = 2(17)
D = 34
27 If and are complementary,Then find the values of y, and .
C Dm Dm C
8, 3 2m C y m B y Start with a labeled diagram.
CD
y - 8
3y + 2
y – 8 + 3y + 2 = 90
4y - 6 = 90
4y = 96
y = 24
divide by 4C = 24 - 8
C = 16
D = 3(24) + 2
D = 72 + 2
D = 74
28
Use the information to find an equation and solve.
Find the measure of an angle that is twice as large as its supplement.
2( ) x = 180 – x
x = 180 – 2x
3x = 180
x = 60
180 – 60 = 120
29Use the information to find an equation and solve.
Find the measure of an angle that is half as large as its complement.
x =1
( )2
90 - x
Multipy by 2 to get rid of fractions
22
2x = 90 - x
3x = 90
x = 30
90 – 30 = 60
30
Use the information to find an equation and solve.
The measure of a supplement of an angle is12 more than twice the measure of the angle.
180 – x = 12 + 2x
180 = 12 + 3x
168 = 3x
56 = x
180 – 56 = 124
31
Use the information to find an equation and solve.
A supplement of an angle is six times as large as the complement of the angle.
180 – x = 6( )90 - x
180 – x = 540 – 6x
180 + 5x = 540
5x = 360
x = 72
Supplement
180 – 72 = 108
Complement
90 – 72 = 18
32 Find the values of x and y.
x
(2y – 17)
(3x – 8)
x + 3x – 8 = 180
4x – 8 = 180
4x = 188
x = 47
47
47 + 2y – 17 = 180
2y + 30 = 180
2y = 150
y = 75
33 Find the values of x and y.
50
x3x - y
2x - 16 2x – 16 = 50
2x = 66
x = 33
33
33 = 3(33) - y
33 = 99 - y
- 66 = - y
66 = y