math 310 section 9.3 more on angles. linear pair def two angles forming a line are called a linear...
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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