Math 310

- Section 9.3
- More on Angles

Linear Pair

- Def
- Two angles forming a line are called a linear

pair.

Ex.

Not a linear pair ltABC ltFDE

Linear pairs ltABC ltDBC ltBDE ltFDE

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.

Vertical angles ltABC ltEBD ltCBE ltDBA

Vertical Angle Theorem

- Thrm
- Vertical angles are congruent.

Ex.

If mltABC 95 find the other three angle

measures.

mltEBD 95 mltCBE 85 mltDBA 85

Supplementary Angles

- Def
- Supplementary angles are any two angles whose sum

of their measures is 180.

Ex.

ltABC ltCBE ltABC ltFED ltABC ltBEG ltDEB ltFED

ltDEB ltCBE ltDEB ltBEG ltGEF ltFED ltGEF ltCBE

ltGEF ltBEG

Given ltABC is congruent to ltFEG Find all pairs

of supplementary angles.

Complementary Angles

- Def
- Complementary angles are any two angles whose sum

of their measures is 90.

Ex.

Given ray BC is perpendicular to line AE. Name

all pairs of complementary angles.

ltCND ltDBE

Ex.

Name all pairs of complementary angles.

ltABC ltGHI ltDEF ltGHI

Transversal

- Def
- A line, crossing two other distinct lines is

called a transversal of those lines.

Ex.

Name two lines and their transversal.

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

ltJKO ltMKO ltQOK ltNOK

Exterior Angles

ltJKL ltMKL ltQOP ltNOP

Alternate Interior Angles

ltJKO ltNOK ltMKO ltQOK

Alternate Exterior Angles

ltJKL ltNOP ltMKL ltQOP

Corresponding Angles

ltJKL ltQOK ltMKL ltNOK ltQOP ltJKO ltNOP ltMKO

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. Name all

congruent angles.

ltABC ltGFB ltDBC ltEFB ltGFH ltABF ltEFH ltDBF

ltABC ltEFH ltDBC ltGFH ltDBF ltGFB ltABF ltEFB

ltABC ltGFB

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