Math 310Math 310

Section 9.3Section 9.3

More on AnglesMore on Angles

Linear PairLinear Pair

DefDef

Two angles forming a line are called a Two angles forming a line are called a linear pairlinear pair..

A

D

C

E

BF

Ex.Ex.

Linear pairs:

<ABC & <DBC <BDE & <FDE

Not a linear pair:

<ABC & <FDE

QuestionQuestion

What can we say about the sum of the What can we say about the sum of the measures of the angles of a linear measures of the angles of a linear pair?pair?

Vertical AnglesVertical Angles

DefDef

When two lines intersect, four angles When two lines intersect, four angles are created. Taking one of the are created. Taking one of the angles, along with the other angle angles, along with the other angle which is which is notnot its linear pair, gives you its linear pair, gives you vertical anglesvertical angles. (ie it is the angle . (ie it is the angle “opposite” of it) “opposite” of it)

Ex.Ex.

A

B

C

DE

Vertical angles:

<ABC & <EBD <CBE & <DBA

Vertical Angle TheoremVertical Angle Theorem

ThrmThrm

Vertical angles are congruent.Vertical angles are congruent.

Ex.Ex.

A

B

C

DE

If m<ABC = 95° find the other three angle measures.

m<EBD = 95°

m<CBE = 85°

m<DBA = 85°

Supplementary AnglesSupplementary Angles

DefDef

Supplementary anglesSupplementary angles are any two are any two angles whose sum of their measures angles whose sum of their measures is 180°.is 180°.

Ex.Ex.

B E

C

A

DF

GGiven: <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

Complementary AnglesComplementary Angles

DefDef

Complementary anglesComplementary angles are any two are any two angles whose sum of their measures angles whose sum of their measures is 90°.is 90°.

Ex.Ex.

A E

D

B

C

Given: ray BC is perpendicular to line AE.

Name all pairs of complementary angles.

<CND & <DBE

Ex.Ex.

B

ACE

D

F

H

I

G65° 65

°

25°

Name all pairs of complementary angles.

<ABC & <GHI <DEF & <GHI

TransversalTransversal

DefDef

A line, crossing two other distinct A line, crossing two other distinct lines is called a lines is called a transversal transversal of those of those lines.lines.

Ex.Ex.

O

KJ M

QN

P

L

Name two lines and their transversal.

Lines: JK & QO Transversal: OK

Transversals and AnglesTransversals and Angles

Given two lines and their transversal, Given two lines and their transversal, two different types of angles are two different types of angles are formed along with 3 different pairs of formed along with 3 different pairs of angles:angles:

Interior anglesInterior angles Exterior anglesExterior angles Alternate interior anglesAlternate interior angles Alternate exterior anglesAlternate exterior angles Corresponding anglesCorresponding angles

Interior AnglesInterior Angles

O

KJ M

QN

P

L

<JKO <MKO <QOK <NOK

Exterior AnglesExterior Angles

O

KJ M

QN

P

L

<JKL <MKL <QOP <NOP

Alternate Interior AnglesAlternate Interior Angles

O

KJ M

QN

P

L

<JKO & <NOK <MKO & <QOK

Alternate Exterior AnglesAlternate Exterior Angles

O

KJ M

QN

P

L

<JKL & <NOP <MKL & <QOP

Corresponding AnglesCorresponding Angles

O

KJ M

QN

P

L

<JKL & <QOK <MKL & <NOK

<QOP & <JKO <NOP & <MKO

Parallel Lines and Parallel Lines and TransversalsTransversals

ThrmThrm

If any two distinct coplanar lines are If any two distinct coplanar lines are cut by a transversal, then a pair of cut by a transversal, then a pair of corresponding angles, alternate corresponding angles, alternate interior angles, or alternate exterior interior angles, or alternate exterior angles are congruent iff the lines are angles are congruent iff the lines are parallel.parallel.

Ex.Ex.

AB

F

CD

EG

H

Given: Lines AB and GF are parallel.

Name all congruent angles.

<ABC & <EFH <DBC & <GFH <DBF & <GFB <ABF & <EFB <ABC & <GFB

<ABC & <GFB <DBC & <EFB <GFH & <ABF <EFH & <DBF

Triangle SumTriangle Sum

ThrmThrm

The sum of the measures of the The sum of the measures of the interior angles of a triangle is 180°.interior angles of a triangle is 180°.

Angle Properties of a Angle Properties of a PolygonPolygon

ThrmThrm The sum of the measures of the The sum of the measures of the

interior angles of any convex interior angles of any convex polygon with polygon with nn sides is sides is 180n – 360180n – 360 or or (n – 2)180(n – 2)180..

The measure of a single interior The measure of a single interior angle of a regular angle of a regular nn-gon is -gon is (180n – (180n – 360)/n360)/n or or (n – 2)180/n(n – 2)180/n..

Ex.Ex.

What is the sum of the interior angles What is the sum of the interior angles of a heptagon? A dodecagon?of a heptagon? A dodecagon?

Heptagon: (7 – 2)180° = (5)180° = Heptagon: (7 – 2)180° = (5)180° = 900°900°

Dodecagon: (10 – 2)180° = (8)180° = Dodecagon: (10 – 2)180° = (8)180° = 1440°1440°

Exterior Angle TheoremExterior Angle Theorem

ThrmThrm

The sum of the measures of the The sum of the measures of the exterior angles (one at each vertex) exterior angles (one at each vertex) of a convex polygon is 360°.of a convex polygon is 360°.

ProofProof

Given a convex polygon with Given a convex polygon with nn sides sides and vertices, lets say the measure of and vertices, lets say the measure of each interior angles is xeach interior angles is x11, x, x22, …., x, …., xnn. . Then the measure of one exterior Then the measure of one exterior angle at each vertices is 180 – xangle at each vertices is 180 – xii. . Adding up all the exterior angles:Adding up all the exterior angles:

(180 – x(180 – x11) + (180 – x) + (180 – x22) + … + (180 – x) + … + (180 – xnn) )

= 180= 180nn – (x – (x11 + x + x22 +…+ x +…+ xnn))

= 180= 180nn – ( – (180n – 360180n – 360 ) )

= 180= 180nn – 180 – 180nn + 360 = 360 + 360 = 360

Ex.Ex.

Pg 610 – 12aPg 610 – 12a

Pg 610 - 7Pg 610 - 7