# 6.2 Properties of Parallelograms

Document Sample

```					6.2 Properties of Parallelograms
Objectives/Assignment

• Use properties of parallelograms
• Assignment: 2-36 even, 55-58 all
In this lesson . . .
And the rest of the chapter, you will study special
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
Q          R
is a parallelogram,
then its opposite sides
are congruent.
►PQ≅RS and SP≅QR

P       S
Q       R
is a parallelogram,
then its opposite
angles are congruent.

P ≅ R and
Q ≅ S
P       S
Q       R
• 6.4—If a quadrilateral is a
parallelogram, then its
consecutive angles are
180°).
mP +mQ = 180°,
mQ +mR = 180°,
mR + mS = 180°,
P       S
mS + mP = 180°
Q       R
is a parallelogram, then
its diagonals bisect
each other.
QM ≅ SM and
PM ≅ RM
P        S
Ex. 1: Using properties of
Parallelograms
5           G
F
•    FGHJ is a
parallelogram. Find the                K   3
unknown length.
a. JH                                              H
J
b. JK
b.
Ex. 1: Using properties of
Parallelograms
5           G
•   FGHJ is a parallelogram.   F
Find the unknown
reasoning.
a.   JH
b.   JK

J                   H
SOLUTION:
a. JH = FG Opposite sides          b.
of a   are ≅.
JH = 5 Substitute 5 for
FG.
Ex. 1: Using properties of
Parallelograms
5               G
•   FGHJ is a parallelogram.   F
Find the unknown
reasoning.
a.   JH
b.   JK

J                               H
SOLUTION:
a. JH = FG Opposite sides      b. b. = GK Diagonals of a
JK
of a   are ≅.                bisect each other.
JH = 5 Substitute 5 for
JK = 3 Substitute 3 for GK
FG.
Ex. 2: Using properties of parallelograms
Q         R
PQRS is a parallelogram.
Find the angle measure.
a. mR
70°
b. mQ                  P             S
Ex. 2: Using properties of parallelograms
Q                    R
PQRS is a parallelogram.
Find the angle measure.
a. mR
70°
b. mQ                  P                          S
a. mR = mP      Opposite angles of a    are ≅.
mR = 70°       Substitute 70° for mP.
Ex. 2: Using properties of parallelograms
Q                          R
PQRS is a parallelogram.
Find the angle measure.
a. mR
70°
b. mQ                  P                               S
a. mR = mP         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.
Ex. 3: Using Algebra with Parallelograms
P                                            Q

PQRS is a parallelogram.
Find the value of x.              S   3x°                120°
R

mS + mR = 180°    Consecutive s of a □ are supplementary.
3x + 120 = 180   Substitute 3x for mS and 120 for mR.
3x = 60   Subtract 120 from each side.
x = 20   Divide each side by 3.
Ex. 4: Proving Facts about Parallelograms  A
E
B
2

Given: ABCD and AEFG are               D                       1
parallelograms.                                                C

Prove 1 ≅ 3.                     G
3
F

1.   ABCD is a □. AEFG is a   1.       Given
▭.
2.   1 ≅ 2, 2 ≅ 3
3.   1 ≅ 3
Ex. 4: Proving Facts about Parallelograms     A
E
B
2

Given: ABCD and AEFG are                  D                        1
parallelograms.                                                    C

Prove 1 ≅ 3.                        G
3
F

1.   ABCD is a □. AEFG is a □.   1.       Given
2.   1 ≅ 2, 2 ≅ 3            2.       Opposite s of a        ▭ are ≅
3.   1 ≅ 3
Ex. 4: Proving Facts about Parallelograms     A
E
B
2

Given: ABCD and AEFG are                  D                        1
parallelograms.                                                    C

Prove 1 ≅ 3.                        G
3
F

1.   ABCD is a □. AEFG is a □.   1.       Given
2.   1 ≅ 2, 2 ≅ 3            2.       Opposite s of a        ▭ are ≅
3.   1 ≅ 3                     3.       Transitive prop. of
congruence.
Ex. 5: Proving Theorem 6.2 A                       B

Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
C
D

1.   ABCD is a .                 1.   Given
2.   Draw BD.
3.   AB ║CD, AD ║ CB.
4.   ABD ≅ CDB, ADB ≅ 
CBD
5.   DB ≅ DB
7.   AB ≅ CD, AD ≅ CB
Ex. 5: Proving Theorem 6.2 A                                  B

Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.                        D
C

1.   ABCD is a .                 1.   Given
2.   Draw BD.                     2.   Through any two points, there
exists exactly one line.
3.   AB ║CD, AD ║ CB.
4.   ABD ≅ CDB, ADB ≅ 
CBD
5.   DB ≅ DB
7.   AB ≅ CD, AD ≅ CB
Ex. 5: Proving Theorem 6.2 A                                  B

Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.                        D
C

1.   ABCD is a .                 1.   Given
2.   Draw BD.                     2.   Through any two points, there
exists exactly one line.
3.   Definition of a parallelogram
3.   AB ║CD, AD ║ CB.
4.   ABD ≅ CDB, ADB ≅ 
CBD
5.   DB ≅ DB
7.   AB ≅ CD, AD ≅ CB
Ex. 5: Proving Theorem 6.2 A                                  B

Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.                        D
C

1.   ABCD is a .                 1.   Given
2.   Draw BD.                     2.   Through any two points, there
exists exactly one line.
3.   Definition of a parallelogram
3.   AB ║CD, AD ║ CB.
4.   Alternate Interior s Thm.
4.   ABD ≅ CDB, ADB ≅ 
CBD
5.   DB ≅ DB
7.   AB ≅ CD, AD ≅ CB
Ex. 5: Proving Theorem 6.2 A                                  B

Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.                        D
C

1.   ABCD is a .                 1.   Given
2.   Draw BD.                     2.   Through any two points, there
exists exactly one line.
3.   Definition of a parallelogram
3.   AB ║CD, AD ║ CB.
4.   Alternate Interior s Thm.
4.   ABD ≅ CDB, ADB ≅ 
CBD
5.   DB ≅ DB                      5.   Reflexive property of congruence
7.   AB ≅ CD, AD ≅ CB
Ex. 5: Proving Theorem 6.2 A                                  B

Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.                        D
C

1.   ABCD is a .                 1.   Given
2.   Draw BD.                     2.   Through any two points, there
exists exactly one line.
3.   Definition of a parallelogram
3.   AB ║CD, AD ║ CB.
4.   Alternate Interior s Thm.
4.   ABD ≅ CDB, ADB ≅ 
CBD
5.   DB ≅ DB                      5.   Reflexive property of congruence
6.   ∆ADB ≅ ∆CBD                  6.   ASA Congruence Postulate
7.   AB ≅ CD, AD ≅ CB
Ex. 5: Proving Theorem 6.2 A                                  B

Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.                        D
C

1.   ABCD is a .                 1.   Given
2.   Draw BD.                     2.   Through any two points, there
exists exactly one line.
3.   Definition of a parallelogram
3.   AB ║CD, AD ║ CB.
4.   Alternate Interior s Thm.
4.   ABD ≅ CDB, ADB ≅ 
CBD
5.   DB ≅ DB                      5.   Reflexive property of congruence
6.   ∆ADB ≅ ∆CBD                  6.   ASA Congruence Postulate
7.   AB ≅ CD, AD ≅ CB             7.   CPOCTAC
Ex. 6: Using parallelograms in real life

FURNITURE DESIGN. A drafting
table is made so that the legs can be       C
joined in different ways to change
the slope of the drawing surface.
In the arrangement below, the legs      B
AC and BD do not bisect each
other. Is ABCD a parallelogram?

A       D
Ex. 6: Using parallelograms in real life

FURNITURE DESIGN. A drafting
table is made so that the legs can be       C
joined in different ways to change
the slope of the drawing surface.
In the arrangement below, the legs      B
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          A       D
would bisect AC. They do not, so
it cannot be a parallelogram.

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 0 posted: 6/8/2013 language: Unknown pages: 26