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Warm-Up #14, Wednesday, 3/9
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2
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Homework
* Triangle Proofs Worksheet #1, 2, 3, 4, 6, 10
Properties of
Parallel Lines
Key Concepts, continued
Theorem
Angles supplementary to the same angle or to congruent
angles are congruent.
If mÐ1+ mÐ2 = 180 and mÐ2 + mÐ3 = 180, then Ð1@ Ð3.
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Theorem
Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a
segment, then that point is equidistant from the
endpoints of the segment.
If a point is equidistant from the endpoints of a
segment, then the point lies on the perpendicular
bisector of the segment.
(continued)
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1.8.1: Proving the Vertical Angles Theorem
Key Concepts, continued
Theorem
If
is the perpendicular bisector of AC, then DA = DC.
If DA = DC, then
is the perpendicular bisector of AC.
1.8.1: Proving the Vertical Angles Theorem
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Midpoint is the middle point of a line
segment. It is equidistant from both
endpoints, and it is the centroid both of the
segment and of the endpoints. It bisects
the segment.
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.
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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.
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4
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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.
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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.
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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.
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
Using the SAS Congruence Postulate
Prove that
 AEB DEC.
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2
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1
Statements
Reasons
AE  DE, BE  CE
Given
1 2
 AEB   DEC
2
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.
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DRA 
4
RA  RG
Given
5
DR  DR
Reflexive Property of Congruence
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 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.
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?