Transcript 6.2 Properties of Parallelograms

6.2 Properties of Parallelograms
In this lesson . . .
And the rest of the chapter, you will study special
quadrilaterals. A parallelogram is a quadrilateral
with both pairs of opposite sides parallel.
When you mark diagrams of quadrilaterals, use
matching arrowheads to indicate which sides are
parallel. For example, in the diagram to the right,
PQ║RS and QR║SP. The symbol
PQRS is
read “parallelogram PQRS.”
FOUR - Theorems about parallelograms
Q
R
• 6.2—If a quadrilateral
is a parallelogram,
then its opposite sides
are congruent.
►PQ≅RS and SP≅QR
P
S
Theorems about parallelograms
Q
R
• 6.3—If a quadrilateral
is a parallelogram,
then its opposite
angles are congruent.
P ≅ R and
Q ≅ S
P
S
Theorems about parallelograms
Q
• 6.4—If a quadrilateral is a
parallelogram, then its
consecutive angles are
supplementary (add up to
180°).
mP +mQ = 180°,
mQ +mR = 180°,
mR + mS = 180°,
P
mS + mP = 180°
R
S
Theorems about parallelograms
Q
R
• 6.5—If a quadrilateral
is a parallelogram, then
its diagonals bisect
each other.
QM ≅ SM and
PM ≅ RM
P
S
Ex. 1: Using properties of
Parallelograms
•
FGHJ is a
parallelogram. Find the
unknown length.
Explain your reasoning.
a. JH
b. JK
5
F
G
K
J
b.
3
H
Ex. 1: Using properties of
Parallelograms
•
FGHJ is a parallelogram.
Find the unknown
length. Explain your
reasoning.
a.
b.
5
F
G
K
3
JH
JK
SOLUTION:
a. JH = FG Opposite sides
of a
are ≅.
JH = 5 Substitute 5 for
FG.
J
b.
H
Ex. 1: Using properties of
Parallelograms
•
FGHJ is a parallelogram.
Find the unknown
length. Explain your
reasoning.
a.
b.
5
F
G
K
3
JH
JK
SOLUTION:
a. JH = FG Opposite sides
of a
are ≅.
JH = 5 Substitute 5 for
FG.
J
b. b.
JK = GK Diagonals of a
bisect each other.
JK = 3 Substitute 3 for GK
H
Ex. 2: Using properties of parallelograms
Q
PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
P
R
70°
S
Ex. 2: Using properties of parallelograms
Q
PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
P
a. mR = mP
mR = 70°
R
70°
Opposite angles of a
are ≅.
Substitute 70° for mP.
S
Ex. 2: Using properties of parallelograms
Q
PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
P
R
70°
S
Opposite angles of a
are ≅.
mR = 70°
Substitute 70° for mP.
b. mQ + mP = 180° Consecutive s of a
are supplementary.
mQ + 70° = 180° Substitute 70° for mP.
mQ = 110° Subtract 70° from each side.
a. mR = mP
Ex. 3: Using Algebra with Parallelograms
P
PQRS is a parallelogram.
Find the value of x.
mS + mR = 180°
3x + 120 = 180
3x = 60
x = 20
S
3x°
120°
R
Consecutive s of a □ are supplementary.
Substitute 3x for mS and 120 for mR.
Subtract 120 from each side.
Divide each side by 3.
Q
Ex. 4: Proving Facts about Parallelograms
A
E
B
2
Given: ABCD and AEFG are
parallelograms.
Prove 1 ≅ 3.
D
1
C
G
3
F
1.
ABCD is a □. AEFG is a
2.
3.
1 ≅ 2, 2 ≅ 3
1 ≅ 3
▭.
1. Given
Ex. 4: Proving Facts about Parallelograms
A
E
B
2
Given: ABCD and AEFG are
parallelograms.
Prove 1 ≅ 3.
D
1
C
G
3
F
1.
2.
3.
ABCD is a □. AEFG is a □.
1 ≅ 2, 2 ≅ 3
1 ≅ 3
1. Given
2.
Opposite s of a
▭ are ≅
Ex. 4: Proving Facts about Parallelograms
A
E
B
2
Given: ABCD and AEFG are
parallelograms.
Prove 1 ≅ 3.
D
1
C
G
3
F
1.
2.
3.
ABCD is a □. AEFG is a □.
1 ≅ 2, 2 ≅ 3
1 ≅ 3
1. Given
▭ are ≅
2.
Opposite s of a
3.
Transitive prop. of
congruence.
Ex. 5: Proving Theorem 6.2 A
B
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
D
1.
2.
3.
4.
5.
6.
7.
ABCD is a .
Draw BD.
AB ║CD, AD ║ CB.
ABD ≅ CDB, ADB ≅ 
CBD
DB ≅ DB
∆ADB ≅ ∆CBD
AB ≅ CD, AD ≅ CB
1.
Given
C
Ex. 5: Proving Theorem 6.2 A
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
1.
2.
ABCD is a .
Draw BD.
3.
4.
AB ║CD, AD ║ CB.
ABD ≅ CDB, ADB ≅ 
CBD
DB ≅ DB
5.
6.
7.
∆ADB ≅ ∆CBD
AB ≅ CD, AD ≅ CB
B
D
1.
2.
Given
Through any two points, there
exists exactly one line.
C
Ex. 5: Proving Theorem 6.2 A
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
D
1.
2.
ABCD is a .
Draw BD.
1.
2.
3.
4.
AB ║CD, AD ║ CB.
ABD ≅ CDB, ADB ≅ 
CBD
DB ≅ DB
3.
5.
6.
7.
∆ADB ≅ ∆CBD
AB ≅ CD, AD ≅ CB
B
Given
Through any two points, there
exists exactly one line.
Definition of a parallelogram
C
Ex. 5: Proving Theorem 6.2 A
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
D
1.
2.
ABCD is a .
Draw BD.
1.
2.
3.
4.
AB ║CD, AD ║ CB.
ABD ≅ CDB, ADB ≅ 
CBD
DB ≅ DB
3.
4.
5.
6.
7.
∆ADB ≅ ∆CBD
AB ≅ CD, AD ≅ CB
B
Given
Through any two points, there
exists exactly one line.
Definition of a parallelogram
Alternate Interior s Thm.
C
Ex. 5: Proving Theorem 6.2 A
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
B
D
C
1.
2.
ABCD is a .
Draw BD.
1.
2.
3.
4.
AB ║CD, AD ║ CB.
ABD ≅ CDB, ADB ≅ 
CBD
DB ≅ DB
3.
4.
Given
Through any two points, there
exists exactly one line.
Definition of a parallelogram
Alternate Interior s Thm.
5.
Reflexive property of congruence
5.
6.
7.
∆ADB ≅ ∆CBD
AB ≅ CD, AD ≅ CB
Ex. 5: Proving Theorem 6.2 A
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
B
D
C
1.
2.
ABCD is a .
Draw BD.
1.
2.
3.
4.
AB ║CD, AD ║ CB.
ABD ≅ CDB, ADB ≅ 
CBD
DB ≅ DB
3.
4.
Given
Through any two points, there
exists exactly one line.
Definition of a parallelogram
Alternate Interior s Thm.
5.
6.
Reflexive property of congruence
ASA Congruence Postulate
5.
6.
7.
∆ADB ≅ ∆CBD
AB ≅ CD, AD ≅ CB
Ex. 5: Proving Theorem 6.2 A
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
B
D
C
1.
2.
ABCD is a .
Draw BD.
1.
2.
3.
4.
AB ║CD, AD ║ CB.
ABD ≅ CDB, ADB ≅ 
CBD
DB ≅ DB
3.
4.
Given
Through any two points, there
exists exactly one line.
Definition of a parallelogram
Alternate Interior s Thm.
5.
6.
7.
Reflexive property of congruence
ASA Congruence Postulate
CPOCTAC
5.
6.
7.
∆ADB ≅ ∆CBD
AB ≅ CD, AD ≅ CB
Ex. 6: Using parallelograms in real life
FURNITURE DESIGN. A drafting
table is made so that the legs can be
joined in different ways to change
the slope of the drawing surface.
In the arrangement below, the legs
AC and BD do not bisect each
other. Is ABCD a parallelogram?
C
B
A
D
Ex. 6: Using parallelograms in real life
FURNITURE DESIGN. A drafting
table is made so that the legs can be
joined in different ways to change
the slope of the drawing surface.
In the arrangement below, the legs
AC and BD do not bisect each
other. Is ABCD a parallelogram?
ANSWER: NO. If ABCD were a
parallelogram, then by Theorem
6.5, AC would bisect BD and BD
would bisect AC. They do not, so
it cannot be a parallelogram.
C
B
A
D