Learning Target
I can use theorems, postulates or definitions to
prove that…
a. vertical angles are congruent.
b. When a transversal crosses parallel lines,
alternate interior angles are congruent and
corresponding angles are congruent, and sameside interior angles are supplementary.
Proving Vertical Angle Theorem
THEOREM
Vertical Angles Theorem
Vertical angles are congruent
1
3,
2
4
Proving Vertical Angle Theorem
GIVEN
PROVE
5 and
6 and
5
6 are a linear pair,
7 are a linear pair
7
Statements
Reasons
1
5 and
6 and
6 are a linear pair,
7 are a linear pair
Given
2
5 and
6 and
6 are supplementary,
7 are supplementary
Linear Pair Postulate
3
5
7
Congruent Supplements Theorem
Third Angles Theorem
The Third Angles Theorem below follows from the Triangle
Sum Theorem.
THEOREM
Third Angles Theorem
If two angles of one triangle are
congruent to two angles of another
triangle, then the third angles are also
congruent.
If  A   D and  B   E, then  C  
F.
Goal 1
PROPERTIES OF PARALLEL LINES
POSTULATE
POSTULATE 15 Corresponding Angles Postulate
If two parallel lines are cut by a transversal,
then the pairs of corresponding angles
are congruent.
1
2
1
2
PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.4 Alternate Interior Angles
If two parallel lines are cut by a transversal,
then the pairs of alternate interior angles are
congruent.
3
4
3
4
PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.5 Consecutive Interior Angles
If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles are
supplementary.
5
6
m
5+m
6 = 180°
PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.6 Alternate Exterior Angles
If two parallel lines are cut by a transversal,
then the pairs of alternate exterior angles are
congruent.
7
8
7
8
PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.7 Perpendicular Transversal
If a transversal is perpendicular to one of two parallel
lines, then it is perpendicular to the other.
j
k
Proving the Alternate Interior Angles Theorem
Prove the Alternate Interior Angles Theorem.
SOLUTION
GIVEN
p || q
PROVE
1
Statements
2
Reasons
1
p || q
Given
2
1
3
3 2
Vertical Angles Theorem
4
1
Transitive property of Congruence
3
2
Corresponding Angles Postulate
Using Properties of Parallel Lines
Given that m 5 = 65°,
find each measure. Tell
which postulate or theorem
you use.
SOLUTION
m
6 = m
5 = 65°
m
7 = 180° – m
m
8 = m
5 = 65°
Corresponding Angles Postulate
m
9 = m
7 = 115°
Alternate Exterior Angles Theorem
Vertical Angles Theorem
5 = 115° Linear Pair Postulate
PROPERTIES OF SPECIAL PAIRS OF ANGLES
Using Properties of Parallel Lines
Use properties of
parallel lines to find
the value of x.
SOLUTION
m
m
4 = 125°
4 + (x + 15)° = 180°
125° + (x + 15)° = 180°
x = 40°
Corresponding Angles Postulate
Linear Pair Postulate
Substitute.
Subtract.
Estimating Earth’s Circumference: History Connection
Over 2000 years ago
Eratosthenes estimated Earth’s
circumference by using the
fact that the Sun’s rays are
parallel.
When the Sun shone exactly
down a vertical well in Syene,
he measured the angle the
Sun’s rays made with a
vertical stick in Alexandria.
He discovered that
m
2
1
50 of a circle
Estimating Earth’s Circumference: History Connection
m
2
1
50 of a circle
Using properties of parallel
lines, he knew that
m
1= m
2
He reasoned that
m
1
1
50 of a circle
Estimating Earth’s Circumference: History Connection
m
1
1
50 of a circle
The distance from Syene to
Alexandria was believed to be
575 miles
1
50 of a circle
Earth’s
circumference
575 miles
Earth’s circumference
50(575 miles)
Use cross product property
29,000 miles
How did Eratosthenes know that m
1=m
2?
Estimating Earth’s Circumference: History Connection
How did Eratosthenes know that m
1=m
SOLUTION
Because the Sun’s rays are parallel,
Angles 1 and 2 are alternate interior
angles, so
1 
2
By the definition of congruent angles,
m
1=m
2
2?
Using the Third Angles Theorem
Find the value of x.
SOLUTIO
N
In the diagram,  N   R and 
L   S.
From the Third Angles Theorem, you know that So, m M = m T.
M  the
T. Triangle Sum Theorem, m M = 180˚– 55˚
From
– 65˚ = 60˚.
Third Angles Theorem
m M = m T
60˚ = (2x + 30)˚
Substitute.
30 = 2x
Subtract 30 from each side.
15 = x
Divide each side by 2.
Example
Learning
Target
Proving Triangles are Congruent
Decide whether the triangles are congruent. Justify your
reasoning.
SOLUTIO
N
Paragraph Proof
From the diagram, you are given that all three corresponding sides are
, NQ and
, QR QM
RP MN PQ
Because P and N have the same measures, P  N.
By the Vertical Angles Theorem, you know that PQR  
NQM.
By the Third Angles Theorem, R  M.
So, all three pairs of corresponding sides and all three
pairs of corresponding angles are congruent. By the
PQR NQM
definition of congruent triangles,

.
Goal 2
Proving Two Triangles are Congruent
Prove that AEB

DEC
.
A
B
E
C
D
GIVEN AB ||DC, AB DC E is the midpoint of BC
,
and AD.
PROVE AEB
DEC.
Plan for Proof Use the fact that  AEB and  DEC are
vertical angles to show that those angles are congruent.
Use the fact that BC intersects parallel segments AB and
DC to identify other pairs of angles that are congruent.
Example
Proving Two Triangles are Congruent
Prove that AEB  DEC
.
A
B
E
SOLUTION
D
C
Statements
Reasons
AB || DC , AB  DC
Given
 EAB   EDC,
 ABE   DCE
Alternate Interior Angles Theorem
 AEB   DEC
Vertical Angles Theorem
E is the midpoint of AD,
E is the midpoint of BC
Given
AE  DE , BE  CE
Definition of midpoint
AEB 
DEC
Definition of congruent triangles
Example
Proving Triangles are Congruent
You have learned to prove that two triangles are congruent by the
definition of congruence – that is, by showing that all pairs of
corresponding angles and corresponding sides are congruent.
THEOREM
B
Theorem 4.4 Properties of Congruent Triangles
A
Reflexive Property of Congruent Triangles
Every triangle is congruent to
itself.
Symmetric Property of Congruent Triangles
D
If ABC  DEF , then

.
DEF ABC
Transitive Property of Congruent Triangles
If ABC  DEF and

, ABC
then JKL J
DEF JKL
C
E
F
L
.
K
Goal 2
Using the SAS Congruence Postulate
Prove that
 AEB DEC.
1
2
3
Stateme
nts
AE  DE, BE  CE
1 2
 AEB   DEC
1
2
Reasons
Given
Vertical Angles Theorem
SAS Congruence Postulate
MODELING A REAL-LIFE SITUATION
Proving Triangles Congruent
ARCHITECTURE You are designing the window shown in the drawing. You
want to make  DRA congruent to  DRG. You design the window so that
DR AG and RA  RG.
Can you conclude that  DRA   DRG ?
D
SOLUTION
GIVEN
PROVE
DR
AG
RA
RG
 DRA
A
 DRG
R
G
Proving Triangles Congruent
GIVEN
PROVE
DR
AG
RA
RG
 DRA
D
 DRG
A
Statements
R
G
Reasons
Given
1
DR
AG
2
DRA and DRG
are right angles.
If 2 lines are , then they form
4 right angles.
3
DRA 
4
RA  RG
Given
5
DR  DR
Reflexive Property of Congruence
6
 DRA   DRG
SAS Congruence Postulate
DRG
Right Angle Congruence Theorem
Congruent Triangles in a Coordinate Plane
Use the SSS Congruence Postulate to show that  ABC   FGH.
SOLUTION
AC = 3 and FH = 3
AC  FH
AB = 5 and FG = 5
AB  FG
Congruent Triangles in a Coordinate Plane
Use the distance formula to find lengths BC and GH.
d=
BC =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(– 4 – (– 7)) 2 + (5 – 0 ) 2
d=
GH =
(x 2 – x1 ) 2 + ( y2 – y1 ) 2
(6 – 1) 2 + (5 – 2 ) 2
=
32 + 52
=
52 + 32
=
34
=
34
Congruent Triangles in a Coordinate Plane
BC = 34 and GH = 34
BC  GH
All three pairs of corresponding sides are congruent,
 ABC   FGH by the SSS Congruence Postulate.