Download Geometry Chap 3 Completed Notes

3-1 Lines and Angles
Parallel Lines – Lines that are coplanar & do not intersect.
__ __ __ __
BF II EJ II DH II CG
Parallel Planes – Planes that do not intersect.
BFG II
EJH
Perpendicular Lines – lines that intersect at a
90 degree angle.
BF I FJ
Skew Lines – lines that are not coplanar, not
parallel, and do not intersect.
Line CG is skew to Line JH
Example A: Refer to the figure.
1) Name all planes that are parallel to plane DEH.
CFG
2) Name all segments that are parallel to AB .
DC II HG II EF
3) Name all segments that intersect with GH .
HD, HE, GC, GF
4) Name all segments that are skew to CD . Lines FG, EH, BF, AE
5) Name a pair of parallel segments.
SR & NM
6) Name a pair of skew segments
MR & KL
7) Name a pair of perpendicular segments
KL & NK
8) Name a pair of parallel planes
RSN & QPK
Transversal – A line that intersects two coplanar
lines at two different points.
Transversal t and the other two lines r and s form 8
angles.
Alternate Interior Angles – nonadjacent angles that lie on opposite sides of the transversal,
and INSIDE (BETWEEN) the other two lines.
<2 & <7; <3 & <6
Alternate Exterior Angles – nonadjacent angles
that lie on opposite sides of the transversal, and OUTSIDE
the other two lines.
<1 & <8; <4 & <5
Corresponding Angles – lie on the same side of the transversal, and on the same sides
of the other two lines
Consecutive (Same Side) Interior Angles –
lie on the same side of the transversal and INSIDE (or between) the other two lines.
Consecutive (Same Side) Exterior Angles –
lie on the same side of the transversal and OUTSIDE the other two lines.
Example B: Use the figure to identify each pair of angles.
1) <1 and <5
Corresponding Angles
2) <9 and <15
Alt. Exterior Angles
3) <2 and <8
Alt. Interior Angles
4) <5 and <15
Alt. Exterior Angles
5) <10 and <14
Corresponding Angles
7) <4 and <7
Same Side Exterior
6) <3 and <10
Same Side Interior
8) <4 and <10
Alt. Interior Angles
3-2 Angles Formed by Parallel Lines and Transversals
When two parallel lines are cut by a transversal, the following pairs of angles are
congruent.
 Corresponding angles
 Alternate Interior angles
 Alternate Exterior angles
When two parallel lines are cut by a transversal, the following pairs of angles are
supplementary.
 Consecutive Exterior angles
 Consecutive Interior angles
Theorems and Postulates You Must Know:
Alternate Interior Angle Theorem: II --- Alt. Int. <’s are congruent
Alternate Exterior Angle Theorem: II --- Alt. Ext. <’s are congruent
Corresponding Angle Postulate: II --- Corresponding <’s are supplementary.
Consecutive Interior Angle Theorem: II --- Same Side Int. <’s are supplementary.
Consecutive Exterior Angle Theorem: II --- Same Side Ext. <’s are supplementary.
Example A:
1) Name the 7 angles that are congruent to <1 .
1 2
5 6
3 4
7 8
3, 6, 8, 9, 11, 14, 15
2) Name the 8 angles that are supplementary to < 11 .
9 10
13 14
11 12
15 16
12, 15, 7, 4, 2, 5, 10, 13
Example B: In the figure, m<3=110, m<10=95. Find the measure of the missing angles.
1) <1
85
2) <13
95
3) <6
85
4) <16
110
5) <2
95
6) <15
70
1 2
5 6
9 10
13 14
3 4
7 8
11 12
15 16
Example C: Find x and y from the figures.
2)
1)
8y+2
10x
3y+1
25y-20
4x-5
3x+11
4x-5 = 3x+11
3y +1 + 4x-5 = 180
10x + 8(6) +2 =180
8y +2 + 25y-20 = 180
x-5 =11
3y +1 +4(16)-5 =180
10x=130
33y -18 =180
x =16
3y +60=180
3y = 120
x =130
y=40
33y =198
y=6
Example C: Find x and the missing angles from the figures.
1)
2)
2x-135=75
2x=210 x=105
x= 105
2x+10=3x-15
10=x-15
25=x
M<ABD=2(25)+10 =60 M<CBD=120
m< ABD = 75
4)
3) Find x and y in the figure.
5x + 4y =55
5x +4(5)=55
25x + 4y =120
-(5x +5y=60)
5x=35
-(25x +5y=125)
Y=5
x=7
Y=5
25x +5(5)=125
25x=100
x=4
3-3 Proving Lines Parallel
More Theorems and Postulates You Must know:
Remember what the Converse is???
Converse Alternate Interior Angle Theorem: Alt. Interior angles are congruent ------- II
Converse Alternate Exterior Angle Theorem: Alt. Exterior angles are congruent ------- II
Converse Corresponding Angle Postulate: Corresponding angles are congruent ------- II
Converse Consecutive Interior Angle Theorem: Same Side Int. angles are supplementary ---- II
Converse Consecutive Exterior Angle Theorem: Same Side Ext. angles are supplementary -- II
Exercises: Determine which lines, if any, are parallel. State the postulate or theorem to justify
your answer.
SSI j II k
Corresponding j II k
Alt. Ext.
l II m
11x -25=7x+35
4x – 25 = 35
4x=60
X=15
X=21
Find the values of x and y.
5x+90=180
4)
5x=90
x=18
6y + 3y=180
5)
x=60
9y=180
Y + 90=180
y=20
Y=90
7)
6)
y + 140 =180
y = 40
2x + 10 =40
2x =30
x=15
Y + 70 + 40=180
y=70
X=70
Find the values of x, y, and z.
8)
9)
x=
y=
z=
x=
y=
z=
10 )
11)
x=
x=
y=
y=
z=
z=
3-4 Perpendicular Lines
Perpendicular Bisector –
The shortest segment from a
to a
segment from the
is the
to the
.
Exercises: Draw the segment that represents the distance indicated.
to
1) ABF
, E to BF
to
2) AHJ
A
E
B
, J to AH
A
F
to
3) ADC
to
, ABC
B
C
H
J
to
4) TVW
, S to VW
T
S
A
W
D
E
V
Exercises: In the figure below, BH AE CF, AE BH, BC BC, CF GD, CE. Name the
segment whose length represents the distance between the following points and lines.
C
B
5) B AE to
6) G
7) C BH to
D
CE to
A
H
G
F
E
8) F to BC
l
m
n
p
q
r
s
t
Exercises: Find x and y.
C
7x-12
(7y-1)°
B 2x + 13
D
Three new theorems to remember:
9)
10)
A
Prove: p
r
3-5 Slope of Lines
Slope/Rate of Change –
Example A: Find the slope of the line that passes through the points.
1) ( 4 ,5) and (2, 1) m =
2) (5,1) and (7, 3)
m=
3) ( 3 ,2) and (3,2)
m=
4) ( 8 ,9)and ( 8
, 8) m =
Positive Slope –
Negative Slope –
Zero Slope –
Undefined –
Parallel Lines –
Perpendicular Lines –
Example B: Determine whether GH and RS are parallel, perpendicular, or neither.
1) G(14,13), H(-11,0), R(-3,7), S(-4,-5)
2) G(15,-9), H(9,-9), R(-4,-1), S(3,-1)
3) G(-4,-10), H(8,14), R(0,4), S(16,-4)
Example C) Find the slope of each line.
1) AB =
6) parallel to AC =
2) AC =
7) perpendicular to AD=
3) AD=
8) perpendicular to AB =
4) CD=
5)
CD=
9) perpendicular to AC =
10) perpendicular to CD=
3-6 Lines in the Coordinate Plane
Slope Intercept Form –
m=
b=
Vertical Line
Horizontal Line
Example A Write the equation of each line with the given information in slope intercept form.
1) Line with slope of 3 through (2,1)
2) line through (0,4) and (-1,2)
3) line with x-intercept 2 and y-intercept 3
4) line through (-3,2) and (1,2)
Example B: Graph the line passing through the given point with the given slope and write
each in slope intercept form.
1) (0,4), m
2) ( 3 ,0), m 0
3) (1,4), undefined
Example C: Graph the line that satisfies the conditions.
1) passes through ( 2 ,2), parallel to
a line whose slope is
2) passes through (0,1), perpendicular
to a line whose slope is
Example D: Graph each line.
1) y
2x 3
2) y
4
3) y
x
Example E: Decide whether the lines are parallel, perpendicular, coincide, or none.
1)
y = 3x + 7
y = –3x – 4
2)
Example F: Write the equation of each line.
b
Line b: m=
b=
Line m : m=
b=
Line r: m=
b=
Line p: m=
b=
m
r
p