Math 310
Section 9.3
More on Angles
Linear Pair
Def
Two angles forming a line are called a linear pair.
Ex.
C
A
D
B
F
Not a linear pair:
<ABC & <FDE
Linear pairs:
<ABC & <DBC
<BDE & <FDE
E
Question
What can we say about the sum of the measures
of the angles of a linear pair?
Vertical Angles
Def
When two lines intersect, four angles are created.
Taking one of the angles, along with the other
angle which is not its linear pair, gives you
vertical angles. (ie it is the angle “opposite” of
it)
Ex.
A
C
B
Vertical angles:
<ABC & <EBD
<CBE & <DBA
D
E
Vertical Angle Theorem
Thrm
Vertical angles are congruent.
Ex.
A
C
B
If m<ABC = 95° find
the other three angle
measures.
D
E
m<EBD = 95°
m<CBE = 85°
m<DBA = 85°
Supplementary Angles
Def
Supplementary angles are any two angles whose
sum of their measures is 180°.
Ex.
C
D
A
B
Given: <ABC is congruent to <FEG
Find all pairs of supplementary angles.
<ABC & <CBE
<ABC & <FED
<ABC & <BEG
<DEB & <FED
<DEB & <CBE
<DEB & <BEG
<GEF & <FED
<GEF & <CBE
<GEF & <BEG
F
E
G
Complementary Angles
Def
Complementary angles are any two angles
whose sum of their measures is 90°.
Ex.
C
A
B
Given: ray BC is perpendicular to line AE.
Name all pairs of complementary angles.
D
E
<CND & <DBE
Ex.
I
C
A
65°
Name all pairs of
complementary
angles.
B
D
E
H
25°
G
65°
F
<ABC & <GHI
<DEF & <GHI
Transversal
Def
A line, crossing two other distinct lines is called a
transversal of those lines.
Ex.
L
J
Q
Name two lines and
their transversal.
K
O
P
M
N
Lines: JK & QO
Transversal: OK
Transversals and Angles
Given two lines and their transversal, two different
types of angles are formed along with 3
different pairs of angles:
 Interior angles
 Exterior angles
 Alternate interior angles
 Alternate exterior angles
 Corresponding angles
Interior Angles
L
J
Q
K
O
P
M
N
<JKO
<MKO
<QOK
<NOK
Exterior Angles
L
J
Q
K
O
P
M
N
<JKL
<MKL
<QOP
<NOP
Alternate Interior Angles
L
J
Q
K
O
P
M
N
<JKO & <NOK
<MKO & <QOK
Alternate Exterior Angles
L
J
Q
K
O
P
M
N
<JKL & <NOP
<MKL & <QOP
Corresponding Angles
L
J
Q
K
O
P
M
N
<JKL & <QOK
<MKL & <NOK
<QOP & <JKO
<NOP & <MKO
Parallel Lines and Transversals
Thrm
If any two distinct coplanar lines are cut by a
transversal, then a pair of corresponding angles,
alternate interior angles, or alternate exterior
angles are congruent iff the lines are parallel.
Ex.
Given: Lines AB and GF are
parallel.
C
Name all congruent angles.
D
B
A
E
F
G
H
<ABC & <EFH
<DBC & <GFH
<DBF & <GFB
<ABF & <EFB
<ABC & <GFB
<ABC & <GFB
<DBC & <EFB
<GFH & <ABF
<EFH & <DBF
Triangle Sum
Thrm
The sum of the measures of the interior angles of
a triangle is 180°.
Angle Properties of a Polygon
Thrm
 The sum of the measures of the interior angles
of any convex polygon with n sides is 180n –
360 or (n – 2)180.
 The measure of a single interior angle of a
regular n-gon is (180n – 360)/n or (n – 2)180/n.
Ex.
What is the sum of the interior angles of a
heptagon? A dodecagon?
Heptagon: (7 – 2)180° = (5)180° = 900°
Dodecagon: (10 – 2)180° = (8)180° = 1440°
Exterior Angle Theorem
Thrm
The sum of the measures of the exterior angles
(one at each vertex) of a convex polygon is
360°.
Proof
Given a convex polygon with n sides and vertices,
lets say the measure of each interior angles is x1,
x2, …., xn. Then the measure of one exterior
angle at each vertices is 180 – xi. Adding up all
the exterior angles:
(180 – x1) + (180 – x2) + … + (180 – xn)
= 180n – (x1 + x2 +…+ xn)
= 180n – (180n – 360 )
= 180n – 180n + 360 = 360
Ex.
Pg 610 – 12a
Pg 610 - 7