Download 1.2 Informal Geometry

1.2 Informal Geometry
Mathematical System:
(Axiomatic System)
• Undefined terms, concepts:
• Point, line, plane, space
• Straightness of a line, flatness of a plane
• A point lies in the interior or the exterior of an angle
• Definitions: give meaning to new terms
• Axioms or Postulates: statements accepted as true
• Theorems: “If P, then Q” need to be proved.
Undefined terms:
• Point
• Line; collinear points
• Plane
• Space
Terms that will be formally defined latter
• Angle
• Triangle
• Rectangle
Line Segment
Definition: A line segment is the part of a line that consists of two end points, and all the points between them.
•
•
•
•
Notation
Measuring
The Midpoint
Congruent line segments
Measuring angles
Congruent angles
Bisect an angle (trisect…)
Straight angle
Right angle
Angles formed by intersection of two lines
Perpendicular lines
Parallel lines (informally)
Empty set
Constructions:
Construct a segment congruent to a given segment
Construct the midpoint of a segment
1.3 Early Definitions and Postulates
Definition: gives meaning to new terms; described precisely a term.
Postulate: a statement assumed to be true
Postulate 1. “Two points determine a line”
• Through two distinct points, there is exactly one line.
Definition:
• The distance between two points A and B is the line segment
AB that joins the two points.
Postulate 2. “Ruler Postulate”
• The measure of any line segment is a unique positive
number.
Definition:
• Congruent line segments are two segments that have the
• same length.
Postulate 3 “Segment -Addition Postulate”
• If X is a point on
and A-X-B, then AX+XB=AB.
Definition:
• The midpoint of a line segment is the point that separates the
line segment into two congruent parts.
Postulate 4
• If two lines intersect, they intersect at a point.
Postulate 5
• If two lines intersect, they intersect at a point.
Postulate 6
• If two distinct planes intersect, then their intersection is
a line.
Postulate 7
• Given two distinct points in a plane, the line containing
these points also lies in the plane.
Definition:
• Ray AB denoted AB is the union of AB and all points X on AB
such that B is between A and X.
Definition:
• Parallel lines are lines that lie in the same plane but do not
intersect.
Definition:
• Coplanar points are points that lie in the same plane.
Theorem 1.3.1
• The midpoint of a line segment is unique
1.4 Angles and their relationships
Definition: An angle is the union of two rays that share a common endpoint.
Postulate 8
• The measure of an angle is a unique positive number.
Types of angles:
• Acute angles
• Right angles
• Obtuse angles
• Straight angles
• Reflex angles
Interior, Exterior, and the sides of an angle.
Postulate 9 “Angle-Addition Postulate”
• If a point D lies in the interior of an angle ABC, then m∠ABD+m∠DBC=m∠ABC.
Definition: Two angles are adjacent (adj.∠s) if they have a common vertex and a common side between them.
Definition: Two angles are congruent (≅∠s) if they have the same measure.
Definition: The bisector of an angle is the ray that separate the given angle into two congruent angles.
Definition: Two angles are complementary if the sum of their measures is 90°
Definition: Two angles are supplementary if the sum of their measures is 180° .
Vertical Angles
Construct an angle congruent to a given angle.
Construct the bisector for a given angle.
1.5 Introduction to geometric proof

Tobelievecertaingeometricprinciples,itisnecessarytohaveproof
Twocolumnproof:
1.
Statement: Statethetheoremtobeproved
2.
Drawing:Representsthehypothesisofthetheorem
3.
Given:Describesthedrowningaccordingtotheinformationfoundinthe
hypothesisofthetheorem
4.
Prove:Describesthedrowningaccordingtotheclaimfoundintheconclusion
ofthetheorem
5.
Proof:Ordersinalogicalflow,alistofclaims(Statements)andjustifications
(Reasons),beginningwiththeGivenandendingwiththeProve
SelectedpropertiesfromAlgebraareusedasreasonstojustifystatements:
Properties of equality (a, b, and c are real numbers)
Addition Property of Equality
If = , then + = +
Subtraction Property of Equality
If = , then − = −
Multiplication Property of Equality
If = , then
=
Division Property of Equality
If = , then ÷ = ÷
Reflexive Property
=
Symmetric Property
If = , then =
Distributive Property
+ =
+
Substitution Property
If = , then replaces b in any equation
Transitive Property
If = , and = then =
Properties of inequality (a, b, and c are real numbers)
Addition Property of Inequality
If > , then + > +
(<)
Subtraction Property of Inequality
If > , then − > −
(<)
Algebraic Proof
Ex:
Given: 2 − 1 = 3
Prove:
=2
Proof
Reasons
Statements
2 −1=3
1.
1. Given
2 −1+1= 3+1
2.
3.
2 =4
4.
=
5.
=2
2. Addition Property of Equality
3. Substitution
4. Division Property of Equality
5. Substitution
GeometricProof
GivenPonthesegment
,provethat
=
−
Drawing
Given: ∈
Prove:
=
A
−
Proof
Statements
1.
Reasons
∈
1. Given
2.
+
=
2. Segment Addition – Postulate 3
3.
=
−
3. Subtraction property of addition
P
B
1.6 Perpendicular Lines
Definition: Perpendicularlinesaretwolinesthatmeettoformcongruentadjacentangles.
Theorem 1.6.1
Iftwolinesareperpendicular,thentheymeettoformrightangles.
Drawing
Given:
Prove:∠
⊥
A
intersecting at E
isarightangle
C
E
D
Proof
Statements
1. Given
⊥
1.
2.
Reasons
∠
≌∠
2. Definition:
3.
∠
=
∠
4.
∠
= 180°
3.Two congruent angles have equal measurements
4. Measure of a straight angle
5.
∠
+
∠
=
6.
∠
+
∠
= 180°
6. Substitution
7.
∠
+
∠
= 180°
7. Substitution
2 ∠
8.
9.
10.
∠
∠
∠
= 180°
= 90°
is a right angle
5. Angle addition postulate
8 Substitution
9. Division Property
10. Definition of right angle
B
Theorem 1.6.2
Iftwolinesintersect,thentheverticalanglesformedarecongruent.
Given:
intersecting
Prove:∠2≌ ∠4
at O
B
Drawing
3
A
O
2
4
1
D
Proof
Statements
1.
2.
intersects
∠
= 180°
3.
∠
4.
∠1 + ∠4 = ∠
∠1 + ∠2 = ∠
5.
∠1 +
1. Given
at O
∠
=
Reasons
∠
∠4 =
2. Definition: measure of straight angles
3. Substitution
∠1 +
∠2
and
4. Angle addition Postulate
∠2
6. Substitution
6.
∠4 =
8.
∠4 ≌ ∠2
8. If two angles are equal in measure the angles are congruent
9.
∠2 ≌ ∠4
9. Symmetric Property of Congruence of angles
7. Subtraction Property
C
Construction 1.
Construct a perpendicular to a given line at a specified point.
Theorem 1.6.3 (NOProof)
Inaplane,thereisexactlyonelineperpendiculartoagivenlineatanypointontheline.
Construction 2.
Construct a perpendicular bisector to a line segment.
1.7 Geometric Proof of a Theorem
Theorems: Statements that can be proved
Hypothesis: given
Conclusion: what we need to establish
Converse of a Theorem
The converse of the statement “If P, then Q” is “If Q, then P”. Interchange the hypothesis with the conclusion.
Exemple:
Theorem 1.6.1
Iftwolinesareperpendicular,thentheymeettoformrightangles.
The converse:
Theorem 1.7.1
Iftwolinesmeettoformrightangles,thentheyareperpendicularlines.
ProofofTheorem 1.7.1
Theorem 1.7.1
Iftwolinesmeettoformrightangles,thentheyareperpendicularlines.
Drawing
Given:
Prove:
and
⊥
intersecting at E so that ∠
C
A
isarightangle
E
B
D
Proof
Statements
1.
and
intersecting at E so that ∠
∠
2.
3.
4.
∠
5. 90° +
6.
7.
8.
9.
+
∠
∠
∠
isarightangle
= 90°
=
∠
= 180
≅
⊥
= 180°
3. If an ∠ is a straight angle, its measure is 180°
4. Angle addition postulate
(2), (3), (4)
6. Subtraction Property of =
=
∠
∠
5. Substitution
= 90°
∠
1. Given
2. If an ∠ is a right angle, its measure is 90°
is a straight ∠, so
∠
Reasons
∠
∠
7. Substitution
(5)
(2), (^)
8. If two ∠ have = measures, the ∠’s are ≌
9. If two lines form ≌ adjacent ∠’s the lines are ⊥
ProofofTheorem 1.7.2
Theorem 1.7.2
Iftwoanglesarecomplementarytothesameangle(ortocongruentangles),thentheseanglesarecongruent.
Given:∠1 and∠3arecomplementary
∠2 and∠4 arecomplementary
∠1 = ∠2
Drawing
Prove:∠3 ≅ ∠4
3
1
Plan:∠1 and∠3=90°
∠2 and∠4=90°
∠1 + ∠3 = ∠2 + ∠4
∠3 = ∠4
4
Proof
Reasons
Statements
1. ∠1 and∠3 arecomplementary,∠2 and∠4arecomplem.
1. Given
2. m∠1 +
2. Defn of complementary angles
3.
∠3=90°and ∠2 +
m∠1 +
∠3= ∠2 +
4.
∠1 =
5.
m∠1 +
6.
∠3 =
7.∠3 ≅ ∠4
∠4=90°.
∠4
3. Substitution
∠2
∠3= ∠1 +
∠4
( 2)
4. Given
∠4
5. Substitution, (3), (4)
6. Subtraction Property of =
7. Defn
(5)
2
Theorem 1.7.3
Iftwoanglesaresupplementarytothesameangle(ortocongruentangles),thentheseanglesarecongruent.
Theorem 1.7.4
Anytworightanglesarecongruent.
Theorem 1.7.5
Iftheexteriorsidesoftwoadjacentacuteanglesformperpendicularrays,thentheseanglesarecomplementary.
Picture Proof of Theorem 1.7.5
Given:
⊥
Prove:∠1 and∠2 arecomplementary
Drawing
A
Proof:With
⊥
,weseethat∠1 and∠2 arepartsof
arightangle.
Thenm∠1 + m∠2 = 90°,so∠1 and∠2 arecomplementary.
D
1
2
B
C
ProofofTheorem 1.7.6
Theorem 1.7.6
Iftheexteriorsidesoftwoadjacentacuteanglesformastraightline,thentheseanglesaresupplementary.
Given:∠3 and∠4and
Drawing
Prove:∠3 and ∠4 are supplementary
H
∙
Plan:
3
E∙
∙F
4
Proof
Reasons
Statements
1. Given
1. ∠3 and∠4 and
2. m∠3 +
∠4 =
∠
2. Angle - Addition Postulate
3. ∠
isastraightangle
3. Defn of a straight angle
4.
∠
4. The measure of a straight angle
5.
m∠3 +
6.
=180°
∠4=180°
∠3 and ∠4 are supplementary
5. Substitution, (2), (4)
6. Defn of supplementary angles
Theorem 1.7.7
Iftwolinesegmentsarecongruent,thentheirmidpointseparatethesesegmentsintofourcongruentsegments.
Theorem 1.7.8
Iftwoanglesarecongruent,thentheirbisectorseparatetheseanglesintofourcongruentangles.
G∙
Chapter 2
2.1:TheParallelPostulate
2.2:IndirectProof
2.3:ProvingLinesParallel
2.4:TheAnglesofaTriangle
2.1:TheParallelPostulate
1. PerpendicularLines
2. ParallelLines
3. EuclidianGeometry
4. ParallelLinesandCongruentAngles
5. ParallelLinesandCongruentAngles
1. PerpendicularLines
Recall:
A. Definition:Twolinesareperpendiculariftheymeettoformcongruentadjacentangles
B. Theorem1.6.1:Perpendicularlinesmeettoformrightangles
C. Construction5:Constructaperpendicularlinetoagivenlineatagivenpoint
D. Constructabisectortoagivenline
Construction6:Constructaperpendicularlinetoagivenlinefromapointnotonthegivenline
Theorem 2.1.1
Fromapointnotonagivenline,thereisexactlyonelineperpendiculartothegivenline.
2.ParallelLines
ParallelLinesandplanes
Definition:Parallellinesarelinesinthesameplanethatdonotintersect
3.EuclidianGeometry
theplaneisflat,twodimensionalsurfaceinwhichthelinesegmentjoininganytwopointsofthe
planeliesentirelywithintheplane
Postulate 10
Throughapointnotonaline,exactlyonelineisparalleltothegivenline.
SpecialAngles
Transversal:alinethatintersecttwo(ormore)otherlinesatdistinctpoints
Interiorangles:Anglesformedbetweenlinem andn:
∠3, ∠4, ∠5, ∠6
Exteriorangles:Anglesformedoutsidelinem andn:
∠1, ∠2, ∠7, ∠8
l
1
3
Correspondingangles:Anglesthatlieinthesamerelativeposition:
left,right,above,below
∠1 and ∠5
∠3 and ∠7
above, left
below, left
∠2 and ∠6
∠4 and ∠8
above, right
below, right
AlternateInteriorAngles:interioranglesthathavedifferent
verticesandlieonoppositesidesofthetransversal:
AlternateExteriorAngles:exterioranglesthathavedifferent
verticesandlieonoppositesidesofthetransversal:
5
7
4
6
8
∠3 and ∠6
∠3 and ∠7
∠1 and ∠8
∠2 and ∠7
2
m
n
Postulate 11
Iftwoparallellinesarecutbyatransversal,thenthepairsofcorrespondinganglesarecongruent.
Correspondingangles:
l
1
∠1 and ∠5, ∠3 and ∠7,∠2 and ∠6, ∠4 and ∠8
Ex:If∠1 = 110° and
3
5
∥ , find ∠2,∠4, ∠5, and ∠8
7
2
m
4
n
6
8
Theorem 2.1.2
Iftwoparallellinesarecutbyatransversalline,thenthepairsofalternateinterioranglesarecongruent.
Given:
l
∥ , transversal l
Prove:∠3 ≅ ∠6
1
3
5
7
6
8
Proof
Reasons
Statements
1.
∥ , transversal l
1. Given
2. ∠2 ≅ ∠6
2 Postulate 11
3. ∠3 ≅ ∠2
3. Vertical Angles, Theorem 1.6.2
4. ∠3 ≅ ∠6
4. Substitution
4
2
m
n
Theorem 2.1.3
Iftwoparallellinesarecutbyatransversal,thenthepairsofalternateexterioranglesarecongruent.
Theorem 2.1.4
Iftwoparallellinesarecutbyatransversal,thenthepairsofinterioranglesonthesamesideofthetransversalaresupplementary.
T
Given:
∥
W
, transversal
Prove:∠1 and ∠3 are supplementary
R
2
U
3
1
Y
V
Proof
Reasons
Statements
1.
∥
, transversal
2. ∠1 ≅ ∠2
3. m∠1 =
4.
m∠
1. Given
2. Alternate interior angles
∠2
3. Defn of congruent angles
= 180°
4. Straight angle
5. m∠2 +
∠3 =
6. m∠2 +
∠3 = 180°
6. Substitution, (4) and (5)
7. m∠1 +
∠3 = 180°
7. Substitution, (3) and (6)
∠1 and ∠3 are supplementary
8. Definition Suppl angles
8.
∠
5. Angle Addition Postulate
X
S
Theorem 2.1.5
Iftwoparallellinearecutbyatransversal,thenthepairsofexterioranglesonthesamesideofthetransversalaresupplementary.
2.2 Indirect Proof
Let →
representstheconditionalstatement“IfP thenQ” and~ =notP.
Thefollowingstatementsarerelatedtothegivenstatement
Conditional →
IfP thenQ
Converse →
IfQ thenP
Inverse~ → ~
IfnotP thennotQ
Contrapositive~ → ~
IfnotQ then notP
Example:IfJuanlivesinLA,thenhelivesinCA
Conditional::IfJuanlivesinLA,thenhelivesinCA
Converse:IfJuanlivesinCA,thenhelivesinLA
Inverse:
IfJuandoesn’tliveinLA,thenhedoesn’tlivesinCA
Contrapositive:IfJuandoesn’tlivesinCA,thenhedoesn’tliveinLA
IndirectProof
→
Given:P
Prove : Q
:
1. Supposethat~Q istrue.
2. Reasonfromthesuppositionuntilyoureachacontradiction
3. Notethatthesuppositionclaimingthat~ istruemustbefalseandthatQ mustthereforebetrue
Ex1: Given
isnotperpendicularto
provethat∠1 and∠2 arenotcomplementaryangles
A
∙
Paragraph Proof: Writinglikeanessay;stilleachstatementhastobejustified.
Supposethat∠1 and∠2 are complementaryangles.
Then m∠1 + ∠2 = 90° because the sum of the measures of two compl. angles is 90°.
Wealsoknowthatm∠1 + ∠2 = ∠
bytheAngle Addition Postulate.
In turn, ∠
=90° by substitution.
Then ∠
isarightangle.
Thus,
⊥
.
But this contradicts the given hypothesis;
therefore, the supposition must be false,
And it follows that ∠1 and∠2 arenotcomplementaryangles.
Statements
C
B
∙
∙
1
2
D
∙
Reasons
1.Supposethat∠1 and∠2 are complementaryangles
1. Contradict the hypothesis
2. m∠1 +
∠2 = 90°
2. The sum of the measures of two compl. angles is 90°.
3. m∠1 +
∠2 =
3. Angle Addition Postulate
4.
∠
5. ∠
6.
∠
=90°
isarightangle
⊥
4. substitution
5. Definition of right angle
6. Theorem of perpendicular lines
7. Contradiction
7. Given
8. ∠1 and∠2 arenotcomplementaryangles
8. Because of contradiction
isnotperpendicularto
Ex2: Completeaformalproofofthefollowingtheorem:
“Iftwolinesarecutbyatransversalsothatthecorrespondinganglesarenotcongruent,thenthetwolinesarenotparallellines”
Given: m andl arecutbythetransversalt and∠1 ≇ ∠5
Prove: ∦m
t
1
Assumethat ∥ .
Whentheselinesarecutbytransversalt, anytwocorrespondingangles(including
∠1 and∠5 arecongruent.But∠1 ≇ ∠5 byhypothesis.Thus,theassumedstatement,
whichclaimsthat ∥ mustbefalse.Itfollowsthat ∦m.
3
l
2
4
B
5
7
6
m
8
Ex3: Completeaformalproofofthefollowingtheorem:
“Thebisectorofanangleisunique”
Given:
bisect∠
Prove:
istheonlybisectorfor∠
A
D
B
Assumethat:
isalsoabisectorof∠
andthatm∠
=
∠
.Giventhat
C
bisect∠
,itfollowsthatm∠
=
∠
.BytheAngleAddition
Postulate,m∠
= ∠
+ m∠
.Bysubstitution,
∠
=
∠
+ ∠
;butthen ∠
= 0,bysubtraction.Anangle
withmeasure0contradictstheProtractorPostulate,therefore,theassumed
statementmustbefalse,anditfollowsthatthebisectorofanangleisunique.
A
E
D
B
C
2.3 Proving line parallel
Recallfrom2.1therelevantPostulateandTheorems:
Postulate 11
Iftwoparallellinesarecutbyatransversal,thenthepairsofcorrespondinganglesarecongruent.
Theorem 2.1.2
Iftwoparallellinesarecutbyatransversalline,thenthepairsofalternateinterioranglesarecongruent.
Theorem 2.1.3
Iftwoparallellinesarecutbyatransversal,thenthepairsofalternateexterioranglesarecongruent.
Theorem 2.1.4
Iftwoparallellinesarecutbyatransversal,thenthepairsofinterioranglesonthesamesideofthetransversalaresupplementary.
Theorem 2.1.5
Iftwoparallellinearecutbyatransversal,thenthepairsofexterioranglesonthesamesideofthetransversalaresupplementary.
Note: ThePostulateandeachTheoremhas
1. Thesamehypothesis:Iftwoparallellinearecutbyatransversal,and
2. Aconclusioninvolvinganglerelationship.
IfwewishtoprovethattwolinesareparallelinsteadofarelationbetweenangleswecanuseConverse.
1. Ahypothesis:ananglerelationship ,and
2. Aconclusion:,thenthetwolinesareparallellines.
Theorem 2.3.1
Iftwolinesarecutbyatransversalsuchthatthepairsofcorrespondinganglesarecongruent,thentheselinesareparallellines.
Theorem 2.3.2
Iftwolinesarecutbyatransversalsuchthatthepairsofthealternateinterioranglesarecongruent,thentheselinesareparallel
lines.
Theorem 2.3.3
Iftwolinesarecutbyatransversalsuchthatthepairsofthealternateexterioranglesarecongruent,thentheselinesareparallel
lines.
Theorem 2.3.4
Iftwolinesarecutbyatransversalsuchthatthealternateinterioranglesonthesamesideofthetransversalaresupplementary,
thentheselinesareparallellines.
Theorem 2.3.5
Iftwolinesarecutbyatransversalsuchthatthealternateexterioranglesonthesamesideofthetransversalaresupplementary,
thentheselinesareparallellines.
Theorem 2.3.1
Iftwolinesarecutbyatransversalsuchthatthepairsofcorrespondinganglesarecongruent,thentheselinesareparallellines.
Given: and
cut by transversal ;∠1 ≅ ∠2
r
Prove: ∥
1
4
t
3
l
Proof:
2
Supposethat ∦m.Thenaliner canbedrownthroughpointP thatisparalleltom;
thisfollowsfromtheParallelPostulate.If ∥ ,then∠3 ≌ ∠2 becausetheseangles
correspond.But∠1 ≌ ∠2 byhypothesis.Now∠3 ≌ ∠1 bytheTransitivePropertyof
Congruence;therefore,m∠3 = ∠1.Butm∠3 + ∠4 = ∠1 (seefig.).Substituting
∠1 for ∠3 leadstom∠1 + ∠4 = ∠1,andbysubtraction, ∠4 = 0. This
contradictstheProtractorPostulate,thereforer andl mustcoincide,anditfollows
that ∥ .
m
Theorem 2.3.2
Iftwolinesarecutbyatransversalsothatthepairsofthealternateinterioranglesarecongruent,thentheselinesareparallel.
Given: and
cut by transversal ;∠2 ≅ ∠3
t
Prove: ∥
l
1
ParagraphProof:
3
2
Proof
Reasons
Statements
1.
and
andtransversal
∠2 ≅ ∠3
1. Given
2.
∠1 ≅ ∠3
2. If two lines intersect, vertical angles are congruent
3.
∠2 ≅ ∠1
3. Transitive property
4.
∥
4. Theorem 2.3.1
m
Theorem 2.3.3
Iftwolinesarecutbyatransversalsuchthatthepairsofthealternateexterioranglesarecongruent,thentheselinesareparallel
lines.
Theorem 2.3.4
Iftwolinesarecutbyatransversalsuchthattheinterioranglesonthesamesideofthetransversalaresupplementary,thenthese
linesareparallellines.
Given: and
cut by transversal ;∠1 issupplementaryto∠2
t
Prove: ∥
l
3
1
2
m
Proof
Statements
1.
2.
and
andtransversal ; ∠1issupplementaryto∠2
∠1 issupplementaryto∠3
∠2 ≅ ∠3
3.
4.
∥
Reasons
1. Given
2. Straight line
3. If two angles are supplementary to the same angle they are ≅
4. Theorem 2.3.1
Theorem 2.3.5
Iftwolinesarecutbyatransversalsuchthattheexterioranglesonthesamesideofthetransversalaresupplementary,thenthese
linesareparallellines.
Theorem 2.3.6
Iftwolinesareeachparalleltoathirdline,thentheselinesareparalleltoeachother.
Theorem 2.3.6
Iftwocoplanarlinesareeachperpendiculartoathirdline,thentheselinesareparalleltoeachother.
Construction 7
Constructthelineparalleltoagivenlinefromapointnotonthatline.
2.4 The Angles of a Triangle
Definition:
Atriangle (symbol∆) is the union of three line segments that are determined by three noncollinear points.
∆
or∆
or∆
or∆
or…
C
,B, C arevertices
aresidesofthetriangle
Points:D apointinsidethetriangle
E pointonthetriangle
F apointoutsidethetriangle
D
.F
A
TrianglesclassifiedbyCongruentsides
C
A
.
.E
B
C
C
B
A
Scalene
Isosceles
B
A
B
Equilateral
TrianglesclassifiedbyCongruentangles
C
A
A
B
B
Acute
C
C
Obtuse
C
B A
A
Equilateral
Right
B
Theorem 2.4.1
Inatrianglethesumofthemeasuresoftheinterioranglesis180°
Given:∆
(a)
C
fig(a)
Prove: ∠ +
∠ + ∠ = 180°
Through C in fig (a), draw
∠1 +
We see that
But
Then
∠1 =
∠ and
∠ +
B
A
Picture Proof
∥
∠2 +
∠3 =
.
∠3 = 180°.
∠
(b)
D
(alternateinteriorangles)
C
1
E
2 3
∠ + ∠ = 180° infig(a)
A
B
Corollary 2.4.2
Eachangleofanequiangulartrianglemeasures60°
Proof:
By Theorem 2.4.1 ∠ + ∠ +m∠ = 180°; Since ∠ = ∠ =m ∠ by hypothesis,
it follows that 3m∠ = 180°. Therefore m∠ = 60°, and ∠ = ∠ =m∠ = 60°;
Corollary 2.4.3
Theacuteanglesofarighttrianglearecomplementary.
Corollary 2.4.4
Iftwoanglesofonetrianglearecongruenttotwoanglesofanothertriangle,thenthethirdanglesarealsocongruent
Corollary 2.4.5
Themeasureofanexteriorangleofatriangleequalsthesumofthemeasuresofthetwononadjacentinteriorangles