Download LP, VA, Parallel Lines

Angle Relationships

CONGRUENT ANGLES: Two angles
are congruent angles if and only if they
have the same measure.
A  B
mA  mB
40o
40o
 A Pair of Complementary Angles – 2 angles whose
measures have the sum of 90o
 A Pair of Supplementary Angles – 2 angles whose
measures have the sum of 180o
Are complementary (/supplementary angles) adjacent angles?

ADJACENT ANGLES: Angles that share a vertex and a side
and whose interiors do not overlap
BAD & CAD
D
Side
NOT BAD & BAC
Vertex
C
Side
 A Pair of Vertical Angles
– angles formed by 2 intersecting lines; they share a
common vertex but not a common side
– If AB and CD intersect at point P so that point P is
between points A and B and also between points C
and D, then APC and BPD are a pair of vertical
angles. APD and BPC are also a pair of vertical
angles.
D
A
P
C
B
 A Linear Pair of Angles
-– adjacent angles whose noncommon sides are
opposite rays
-– If X, Y, Z are consecutive collinear points and W is a
point not on XZ, then XYW and WYZ form a linear
pair of angles.
Are supplementary angles a
linear pair of angles?
W
X
Y
Z
How many linear pair of
angles are formed when 2 lines
intersect?
Theorems

Linear Pair Thm
- If two angles form a linear pair, then they are
supplementary.

Vertical Angles Thm
- If two angles are vertical angles, then they are
congruent.
a=127
a=c=68; b=112
a=c=35;
b=40; d=70
a=b=90; c=42;
d=48; e=132
a=c=20;
b=d=70;
e=110
a=70; b=55;
c=25
Special Angles formed by
a Transversal
Perpendicular Lines () – two lines that
intersect to form a right angle
Parallel Lines (//) – 2 or more lines that are
coplanar and that do not intersect
Skew Lines – lines that are not coplanar and
that do not intersect
Why is ‘coplanar’
not in the
definition of 
lines?
Parallel Lines
Perpendicular Lines

TRANSVERSAL
- A line that intersects 2 or more coplanar lines at
different points

Which of the following has a transversal?
l2
l1
l3
l2
l1
l3
l2
l3
l1
DEFINITION
s 3, 4, 5 & 6 are
INTERIOR ANGLES
5 6
 s 1, 2, 7 & 8 are
7 8
EXTERIOR ANGLES
ALTERNATE interior angles (AIA)
1 2
3 4


- 2 non-adjacent interior angles on opposite
sides of the transversal

Example: s 3 & 6 and s 4 & 5
DEFINITION
s 3, 4, 5 & 6 are
INTERIOR ANGLES
5 6
 s 1, 2, 7 & 8 are
7 8
EXTERIOR ANGLES
ALTERNATE exterior angles (AEA)
1 2
3 4


- 2 non-adjacent exterior angles on opposite
sides of the transversal

Example: s 1 & 8 and s 2 & 7
DEFINITION
1 2
3 4
5 6
7 8


CONSECUTIVE interior
angles (CIA)
- 2 interior angles on the same
side of the transversal
Example: s 3 & 5 and s 4 & 6
DEFINITION
1 2
3 4
5 6
7 8


CORRESPONDING
angles (CA)
- 2 non-adjacent angles on the
same side of the transversal
such that one is an exterior
angle and the other is an
interior angle
Example: s 1 & 5, s 3 & 7, s 2 & 6,
s 4 & 8
Exercise
1
5
9
2
6
3
Identify each angle pair as
AIA, AEA, CA, CIA or none of these.
4
7 8
10
11
12
13 14
15
16
*Identify the transversal .
a. 13 & 5 CA
b. 12 & 7 AIA
c. 10 & 7 none
d. 3 & 1
CA
e. 3 & 16 AEA
f. 13 & 4 none
g. 10 & 1 none
h. 10 & 11 CIA
Parallel Line Properties
PROPERTIES of // Lines

1 2
3 4
5
7
6
8

What if our transversal is
intersecting 2 // lines?
What relationships can
we observe between:
• CA?
• AIA?
• AEA?
• CIA?
congruent
congruent
congruent
supplementary
Parallel Line Theorems
ÌF TWO LINES ARE PARALLEL…

CA Theorem
- …then CORRESPONDING ANGLES are CONGRUENT.

AIA Theorem
- ….then ALTERNATE INTERIOR ANGLES are CONGRUENT.

AEA Theorem
- …then ALTERNATE EXTERIOR ANGLES are
CONGRUENT.

CIA Theorem
- …then CONSECUTIVE INTERIOR ANGLES are
SUPPLEMENTARY. * Prove algebraically.
Practice (Source: DG by Serra)
2.
a=b=c=54
b=d=65; a=c=115
Practice (Source: DG by Serra)
4.
3.
a=72; b=126
Practice
5.
5x + 2 = 182 – 4x
9x = 180
x = 20
182 – 4x  102
102 = 4y + 2
y=25
Practice
6.
7.
Practice
8.
Practice
What’s wrong with this picture? Explain.
9.
Practice
11.
10.
25
y
m
80
m=125
153
115
m=38
HOMEWORK
a=102; b=78;
c=f=58;
d=122; e=26
Parallel Line Properties
(Part II)
PROPERTIES of // Lines

Is the converse of the // Line Thm true?
If 2 lines are cut by a transversal to form
pairs of congruent CA, congruent AIA, and
congruent AEA, then the lines are parallel.
// by the Converse
of AIA Thm
Not // (CIAs are not
supplementary)
What is b so that the 2 lines are parallel?
4x – 12
3x + 2 b
4x – 12 = 3x + 2
x = 14
4x – 12  44
b=136
Not //
Practice
Practice
Determine which lines are parallel.