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6.2 Properties of Parallelograms

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Title: 6.2 Properties of Parallelograms


1
6.2 Properties of Parallelograms
  • Geometry
  • Mrs. Spitz
  • Spring 2005

2
Objectives
  • Use some properties of parallelograms.
  • Use properties of parallelograms in real-lie
    situations such as the drafting table shown in
    example 6.

3
Assignment
  • pp. 333-335 2-37 and 39

4
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, PQRS and QRSP. The symbol PQRS is
    read parallelogram PQRS.

5
Theorems about parallelograms
Q
R
  • 6.2If a quadrilateral is a parallelogram, then
    its opposite sides are congruent.
  • ?PQ?RS and SP?QR

P
S
6
Theorems about parallelograms
Q
R
  • 6.3If a quadrilateral is a parallelogram, then
    its opposite angles are congruent.
  • ?P ? ?R and
  • ?Q ? ?S

P
S
7
Theorems about parallelograms
Q
R
  • 6.4If 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,
  • m?S m?P 180

P
S
8
Theorems about parallelograms
Q
R
  • 6.5If a quadrilateral is a parallelogram, then
    its diagonals bisect each other.
  • QM ? SM and
  • PM ? RM

P
S
9
Ex. 1 Using properties of Parallelograms
5
G
F
  • FGHJ is a parallelogram. Find the unknown
    length. Explain your reasoning.
  • JH
  • JK

3
K
H
J
b.
10
Ex. 1 Using properties of Parallelograms
5
G
F
  • FGHJ is a parallelogram. Find the unknown
    length. Explain your reasoning.
  • JH
  • JK
  • SOLUTION
  • a. JH FG Opposite sides of a are ?.
  • JH 5 Substitute 5 for FG.

3
K
H
J
b.
11
Ex. 1 Using properties of Parallelograms
5
G
F
  • FGHJ is a parallelogram. Find the unknown
    length. Explain your reasoning.
  • JH
  • JK
  • SOLUTION
  • a. JH FG Opposite sides of a are ?.
  • JH 5 Substitute 5 for FG.

3
K
H
J
b.
  • JK GK Diagonals of a bisect each
    other.
  • JK 3 Substitute 3 for GK

12
Ex. 2 Using properties of parallelograms
R
Q
  • PQRS is a parallelogram.
  • Find the angle measure.
  • m?R
  • m?Q

70
P
S
13
Ex. 2 Using properties of parallelograms
R
Q
  • PQRS is a parallelogram.
  • Find the angle measure.
  • m?R
  • m?Q
  • a. m?R m?P Opposite angles of a are ?.
  • m?R 70 Substitute 70 for m?P.

70
P
S
14
Ex. 2 Using properties of parallelograms
R
Q
  • PQRS is a parallelogram.
  • Find the angle measure.
  • m?R
  • m?Q
  • a. m?R m?P Opposite angles of a are ?.
  • m?R 70 Substitute 70 for m?P.
  • 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.

70
P
S
15
Ex. 3 Using Algebra with Parallelograms
P
Q
  • PQRS is a parallelogram. Find the value of x.
  • m?S m?R 180
  • 3x 120 180
  • 3x 60
  • x 20

3x
120
S
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.

16
Ex. 4 Proving Facts about Parallelograms
  • Given ABCD and AEFG are parallelograms.
  • Prove ?1 ? ?3.
  • ABCD is a ?. AEFG is a ?.
  • ?1 ? ?2, ?2 ? ?3
  • ?1 ? ?3
  1. Given

17
Ex. 4 Proving Facts about Parallelograms
  • Given ABCD and AEFG are parallelograms.
  • Prove ?1 ? ?3.
  • ABCD is a ?. AEFG is a ?.
  • ?1 ? ?2, ?2 ? ?3
  • ?1 ? ?3
  1. Given
  2. Opposite ?s of a ? are ?

18
Ex. 4 Proving Facts about Parallelograms
  • Given ABCD and AEFG are parallelograms.
  • Prove ?1 ? ?3.
  • ABCD is a ?. AEFG is a ?.
  • ?1 ? ?2, ?2 ? ?3
  • ?1 ? ?3
  1. Given
  2. Opposite ?s of a ? are ?
  3. Transitive prop. of congruence.

19
Ex. 5 Proving Theorem 6.2
  • Given ABCD is a parallelogram.
  • Prove AB ? CD, AD ? CB.
  • ABCD is a ?.
  • Draw BD.
  • AB CD, AD CB.
  • ?ABD ? ?CDB, ?ADB ? ? CBD
  • DB ? DB
  • ?ADB ? ?CBD
  • AB ? CD, AD ? CB
  1. Given

20
Ex. 5 Proving Theorem 6.2
  • Given ABCD is a parallelogram.
  • Prove AB ? CD, AD ? CB.
  • ABCD is a ?.
  • Draw BD.
  • AB CD, AD CB.
  • ?ABD ? ?CDB, ?ADB ? ? CBD
  • DB ? DB
  • ?ADB ? ?CBD
  • AB ? CD, AD ? CB
  1. Given
  2. Through any two points, there exists exactly one
    line.

21
Ex. 5 Proving Theorem 6.2
  • Given ABCD is a parallelogram.
  • Prove AB ? CD, AD ? CB.
  • ABCD is a ?.
  • Draw BD.
  • AB CD, AD CB.
  • ?ABD ? ?CDB, ?ADB ? ? CBD
  • DB ? DB
  • ?ADB ? ?CBD
  • AB ? CD, AD ? CB
  1. Given
  2. Through any two points, there exists exactly one
    line.
  3. Definition of a parallelogram

22
Ex. 5 Proving Theorem 6.2
  • Given ABCD is a parallelogram.
  • Prove AB ? CD, AD ? CB.
  • ABCD is a ?.
  • Draw BD.
  • AB CD, AD CB.
  • ?ABD ? ?CDB, ?ADB ? ? CBD
  • DB ? DB
  • ?ADB ? ?CBD
  • AB ? CD, AD ? CB
  1. Given
  2. Through any two points, there exists exactly one
    line.
  3. Definition of a parallelogram
  4. Alternate Interior ?s Thm.

23
Ex. 5 Proving Theorem 6.2
  • Given ABCD is a parallelogram.
  • Prove AB ? CD, AD ? CB.
  • ABCD is a ?.
  • Draw BD.
  • AB CD, AD CB.
  • ?ABD ? ?CDB, ?ADB ? ? CBD
  • DB ? DB
  • ?ADB ? ?CBD
  • AB ? CD, AD ? CB
  • Given
  • Through any two points, there exists exactly one
    line.
  • Definition of a parallelogram
  • Alternate Interior ?s Thm.
  • Reflexive property of congruence

24
Ex. 5 Proving Theorem 6.2
  • Given ABCD is a parallelogram.
  • Prove AB ? CD, AD ? CB.
  • ABCD is a ?.
  • Draw BD.
  • AB CD, AD CB.
  • ?ABD ? ?CDB, ?ADB ? ? CBD
  • DB ? DB
  • ?ADB ? ?CBD
  • AB ? CD, AD ? CB
  • Given
  • Through any two points, there exists exactly one
    line.
  • Definition of a parallelogram
  • Alternate Interior ?s Thm.
  • Reflexive property of congruence
  • ASA Congruence Postulate

25
Ex. 5 Proving Theorem 6.2
  • Given ABCD is a parallelogram.
  • Prove AB ? CD, AD ? CB.
  • ABCD is a ?.
  • Draw BD.
  • AB CD, AD CB.
  • ?ABD ? ?CDB, ?ADB ? ? CBD
  • DB ? DB
  • ?ADB ? ?CBD
  • AB ? CD, AD ? CB
  • Given
  • Through any two points, there exists exactly one
    line.
  • Definition of a parallelogram
  • Alternate Interior ?s Thm.
  • Reflexive property of congruence
  • ASA Congruence Postulate
  • CPCTC

26
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?

27
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.
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