chapter 9 parallel lines smpk penabur gading serpong
TRANSCRIPT
Lines A. Position of line
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1 Horizontal line
2 Vertical line 3 Curve line
Lines B. Position of two lines
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1 Intersection lines : two lines that joining in a point
Lines B. Position of two lines
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2 Perpendicular lines : two lines that joining in a point and form 90 degree angle
Lines B. Position of two lines
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3Skew lines : two lines that not intersects and not parallel
Lines B. Position of two lines
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4 Coincide lines : two lines that overlap ( match ) each other
a
b
Lines B. Position of two lines
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4 Parallel lines : two lines not intersects and always have the same distance
a
b
Chapter 3
Parallel Lines
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Vocabulary • Parallel Lines = garis sejajar• Skew lines = garis berpotongan dan tidak
sejajar• Transversal lines = garis yang memotong dua
garis dititik yang berbeda
• Interior angles = sudut dalam• Exterior angles = sudut luar• Corresponding angles = sudut sehadap • Alternate interior angles = sudut dalam berseberangan• Alternate exterior angles = sudut luar berseberangan• Consecutive interior angles = sudut dalam sepihak• Consecutive exterior angles = sudut luar sepihak• Vertical angles = sudut bertolak –belakang
Parallel lines in our daily life
Parallel Lines
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Parallel Lines
Parallel lines in our daily life
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Parallel Lines
Parallel lines in our daily life
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Parallel Lines
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Parallel lines in our daily life
Parallel lin
es in o
ur
daily life
Parallel Lines
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PARALLEL LINES•Def: line that do not intersect and always
have the same distance.
•Illustration:
•Notation: l || m AB || CD
lm
A
B
C
D
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TransversalDefinition: A line that intersects two or more lines in a plane at different points is called a transversal.
When a transversal t intersects line n and m, eight angles are formed.
transversalm
n
1 23 4
5 67 8
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16
INTERIOREXTERIOR
EXTERIOR
m
l
k
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
Angles and Parallel LinesWhen a transversal intersects parallel lines, eight anglesare formed.
transversal1 2
3 4
5 67 8
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Vertical Angles & Linear PairVertical Angles:
Linear Pair:
1 4, 2 3, 5 8, 6 7
Two angles that are opposite angles. Vertical angles are congruent.
1 & 2 , 2 & 4 , 4 &3, 3 & 1, 5 & 6, 6 & 8, 8 & 7, 7 & 5
Supplementary angles that form a line (sum = 180)
1 23 4
5 67 8
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Corresponding Angles
Two angles that occupy corresponding positions.
Top Left
t
Top Left
Top Right
Top Right
Bottom Right
Bottom Right
Bottom Left
Bottom Left
Ð1 5Ð2 6Ð3 7Ð4 8
1 2
3 4
5 6
7 8
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Corresponding Angles
If two parallel lines cut by transversal, so the magnitude of corresponding angles are equal t
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Find the measures of the missing angles
145
?
t
35
145
Corresponding Angles
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Alternate Interior AnglesTwo angles that lie between parallel lines on
opposite sides of the transversal
t
Ð3 6Ð4 5
1 2
3 4
5 6
7 8
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If two parallel lines cut by transversal, so the magnitude of alternate interior angles are equal t
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Alternate Interior Angles
Find the measures of the missing angles
82
?
t
98 82
Alternate Interior Angles
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Two angles that lie outside parallel lines on opposite sides of the transversal
t
Ð2 7Ð1 8
1 2
3 4
5 6
7 8
Alternate Exterior Angles
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If two parallel lines cut by transversal, so the magnitude of alternate exterior angles are equal t
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Alternate Exterior Angles
Alternate Exterior AnglesFind the measures of the missing angles
120
?
t
60 120
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Consecutive ( Allied ) Interior Angles
Two angles that lie between parallel lines on the same sides of the transversal
t
Ð3 +5 = 1800
Ð4 +6 = 1800
1 2
3 4
5 6
7 8
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If two parallel lines cut by transversal, then alternate interior angles are supplementaryt
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Consecutive (Allied) Interior Angles
+ =1800
Consecutive ( Allied ) Interior Angles
Find the measures of the missing angles
?
t
135
45
1800 – 1350 = 450
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Two angles that lie outside parallel lines on the same sides of the transversal.
t
Ð1 + 7 = 1800
Ð2 + 8 = 1800
1 2
3 4
5 6
7 8
Consecutive Exterior Angles
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If two parallel lines cut by transversal, then alternate exterior angles are supplementaryt
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Consecutive (Allied) Exterior Angles
+= 1800
Consecutive Exterior Angles
Find the measures of the missing angles
?
t
117
63
1800 – 1170 = 630
Exercise Find the value of angles number 1 to 7 !
5701
2 3
4 5
6 7
1 = 1800 – 570 = 1230
2 = 570
3 = 1230
4 = 1230
5 = 570
6 = 570
7 = 1230
Parallel Lines
Part II: Equations
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Alternate Exterior Angles
• Name the angle relationship• Are they congruent or supplementary?• Find the value of x
125
t
5x
5x = 125
5 5x = 25
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Corresponding Angles
• Name the angle relationship• Are they congruent or supplementary?• Find the value of x
2x + 1
t
151
2x = 150
2 2x = 75
2x + 1 = 151- 1 - 1
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Consecutive Interior Angles
• Name the angle relationship• Are they congruent or supplementary?• Find the value of x
81
t
7x + 15
supp
7x = 847 7x = 12
7x + 96 = 180 - 96 - 96
7x + 15 + 81 = 180
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Alternate Interior Angles
• Name the angle relationship• Are they congruent or supplementary?• Find the value of x
3x
t
2x + 20
20 = x
2x + 20 = 3x- 2x - 2x
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7X + 30 = 15X -18- 30 -30
7X = 15X - 48-15X -15X
-8X= - 48-8 -8
X = 6
Find the value for X:
Both angles are ALTERNATE EXTERIOR and the lines are parallel, so the angles are equal
7X + 30
15X - 18
14X + 6 = 8X + 54- 6
-614X = 8X + 48-8X -8X6X = 486 6X = 8
Find the value for X:
Both angles are ALTERNATE INTERIOR and the lines are parallel, so the angles are equal
8X + 54
14X + 6
16X + 9 = 9X + 58- 9
-916X = 9X + 49-9X -9X7X = 497 7X = 7
Find the value for X:
Both angles are CORRESPONDING and the lines are parallel, so the angles are equal
9X + 5816X +
9
Find the value for X:
3X + 17
17X + 23
Both angles are CONSECUTIVE INTERIOR ANGLES, so they are SUPPLEMENTARY:
(3X + 17) + (17X + 23) = 180
3X + 17X + 17 + 23 = 180
20X + 40 = 180 -40 -
4020X = 14020 20
X = 7
Both angles are ALTERNATE EXTERIOR :
8X + 26 = 12X -14 - 26 -
268X = 12X - 40-12X -
12X-4X= - 40-4 -4X = 10
=12( ) -14
10= 120 -
14= 106°
Angles form a LINEAR PAIR:
Z + 106° = 180° -106 -106Z = 74
8X + 26
12X – 14
Z
Find the value for X and Z:
Both angles are ALTERNATE INTERIOR :
5Z + 13 = 93 – 3Z - 13 -13
5Z = -3Z + 80+ 3Z + 3Z
8Z = 808 8
Z = 10
= 93 – 3( )
10= 93 - 30= 63°
Y + 63° = 90°-63 -63
Y = 27°
These are complementary:
5Z + 13
93 – 3Z
Y
Find the value for Y and Z in the figure below:
Both angles are ALTERNATE INTERIOR :
6Y + 15 = 75 – 14Y- 15 -15
6Y = -14Y +60+ 14Y +
14Y20Y = 6020 20Y = 3
= 6( ) + 15
3= 18 + 15= 33°
X + 33° = 90°-33 -33
X = 57°
These are complementary:
6Y + 15
75 – 14Y
X
Find the value for X and Y in the figure below:
35° 58°
87°
+ +
C
Cm
B
Bm
A
Am =180°35° + 87° + 58° =
180°The sum of the interior angles of a triangle is always 180°
ANGLE SUM THEOREM:
48
103°77
°
77°
65° 1. Vertical
Angles2. Linear pair:
180°-103° = 77°180°-65°= 115°
115°
115°
3. Corresponding Angles
65°
65°115
°115°
4. Vertical Angles
5. Linear Pair:180°-65°=
115°6. Interior Angle Sum in triangle is 180°:
180°-77°-65° = 38°
38°
7. Vertical Angles
38°
8. Corresponding Angles
38°
38°
9. Linear Pair180°-38°= 142°
142° 142
°142°
142°
103°
65°
Find all the unknown angles in the figure below:
110°70
°
70°
85° 1. Vertical
Angles2. Linear pair:
180°-110° = 70°180°-85°= 95°
95°95°
3. Corresponding Angles
85°
85°95° 95°
4. Vertical Angles
5. Linear Pair:180°-85°=
95°6. Interior Angle Sum in triangle is 180°:
180°-70°-85° = 25°
25°
7. Vertical Angles
25°
8. Corresponding Angles
25°
25°
9. Linear Pair180°-25°= 155°
155° 155
°155°
155°
110°
85°
Find all the unknown angles in the figure below:
Alternate Exterior Angles:Z =
145°
2X + 5
5Y + 5
Z
145°
Find the value for X, Y and Z in the figure below:
Alternate Exterior Angles:Z =
145°Linear Pair and supplementary:
145° + (5Y + 5)° = 180° 150 + 5Y =
180-150 -150
5Y = 305 5
Y = 6
2X + 5
5Y + 5
Z
145°
Find the value for X, Y and Z in the figure below:
Alternate Exterior Angles:Z =
145°Linear Pair and supplementary:
145° + (5Y + 5)° = 180° 150 + 5Y =
180-150 -150
5Y = 305 5
Y = 6Corresponding angles:2X + 5 = 145°-5 -
52X = 1402 2X = 70
2X + 5
5Y + 5
Z
145°
Find the value for X, Y and Z in the figure below:
Alternate Exterior Angles:T =
140°
2R – 15
4S – 20
T
140°
Find the value for R, S and T in the figure below:
Alternate Exterior Angles:T =
140°Linear Pair and supplementary:
140° + (4S – 20 )° = 180° 120 + 4S =
180-120 -120
4S = 604 4
S = 15
2R – 15
4S – 20
T
140°
Find the value for R, S and T in the figure below:
Alternate Exterior Angles:Z =
140°Linear Pair and supplementary:
140° + (4S – 20 )° = 180°
120 + 4S = 180-120 -
1204S = 604 4
S = 15
Corresponding angles:2R – 15 = 140°+15
+152R = 1552 2R = 77.5
2R – 15
4S – 20
T
140°
Find the value for R, S and T in the figure below:
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