6.2 Properties of Parallelograms

Document Sample
6.2 Properties of Parallelograms Powered By Docstoc
					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
 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       R
• 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       S
    mS + mP = 180°
 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
                                        5           G
                               F
•    FGHJ is a
     parallelogram. Find the                K   3
     unknown length.
     Explain your reasoning.
     a. JH                                              H
                                    J
     b. JK
                                   b.
Ex. 1: Using properties of
Parallelograms
                                        5           G
•   FGHJ is a parallelogram.   F
    Find the unknown
    length. Explain your                    K   3
    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
    length. Explain your                            K    3
    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
6.   ∆ADB ≅ ∆CBD
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
6.   ∆ADB ≅ ∆CBD
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
6.   ∆ADB ≅ ∆CBD
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
6.   ∆ADB ≅ ∆CBD
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
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