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

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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.

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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
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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
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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
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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.
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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.
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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

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Ex. 2 Using properties of parallelograms
R
Q

• PQRS is a parallelogram.
• Find the angle measure.
• m?R
• m?Q

70
P
S
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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
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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
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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.

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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

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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 ?

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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.

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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

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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.

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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

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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.

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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

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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

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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

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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?

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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.