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6.2 Properties of Parallelograms. Geometry Mrs. Spitz Spring 2005. Objectives:. Use some properties of parallelograms. Use properties of parallelograms in real-lie situations such as the drafting table shown in example 6. Assignment:. pp. 333-335 #2-37 and 39. In this lesson . . . .

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6.2 Properties of Parallelograms


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6 2 properties of parallelograms

6.2 Properties of Parallelograms

Geometry

Mrs. Spitz

Spring 2005

objectives
Objectives:
  • Use some properties of parallelograms.
  • Use properties of parallelograms in real-lie situations such as the drafting table shown in example 6.
assignment
Assignment:
  • pp. 333-335 #2-37 and 39
in this lesson
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.”

theorems about parallelograms
6.2—If a quadrilateral is a parallelogram, then its opposite sides are congruent.

►PQ≅RS and SP≅QR

Theorems about parallelograms

Q

R

P

S

theorems about parallelograms6
6.3—If a quadrilateral is a parallelogram, then its opposite angles are congruent.

P ≅ R and

Q ≅ S

Theorems about parallelograms

Q

R

P

S

theorems about parallelograms7
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°,

mS + mP = 180°

Theorems about parallelograms

Q

R

P

S

theorems about parallelograms8
6.5—If a quadrilateral is a parallelogram, then its diagonals bisect each other.

QM ≅ SM and

PM ≅ RM

Theorems about parallelograms

Q

R

P

S

ex 1 using properties of parallelograms10
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.

Ex. 1: Using properties of Parallelograms

5

G

F

3

K

H

J

b.

ex 1 using properties of parallelograms11
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.

Ex. 1: Using properties of Parallelograms

5

G

F

3

K

H

J

b.

  • JK = GK Diagonals of a bisect each other.
  • JK = 3 Substitute 3 for GK
ex 2 using properties of parallelograms13
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.

Ex. 2: Using properties of parallelograms

Q

R

70°

P

S

ex 2 using properties of parallelograms14
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.

Ex. 2: Using properties of parallelograms

Q

R

70°

P

S

ex 3 using algebra with parallelograms
PQRS is a parallelogram. Find the value of x.

mS + mR = 180°

3x + 120 = 180

3x = 60

x = 20

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.

Ex. 3: Using Algebra with Parallelograms

P

Q

3x°

120°

S

R

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

Given

Ex. 4: Proving Facts about Parallelograms
ex 4 proving facts about parallelograms17
Given: ABCD and AEFG are parallelograms.

Prove 1 ≅ 3.

ABCD is a □. AEFG is a □.

1 ≅ 2, 2 ≅ 3

1 ≅ 3

Given

Opposite s of a▭ are ≅

Ex. 4: Proving Facts about Parallelograms
ex 4 proving facts about parallelograms18
Given: ABCD and AEFG are parallelograms.

Prove 1 ≅ 3.

ABCD is a □. AEFG is a □.

1 ≅ 2, 2 ≅ 3

1 ≅ 3

Given

Opposite s of a▭ are ≅

Transitive prop. of congruence.

Ex. 4: Proving Facts about Parallelograms
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

Ex. 5: Proving Theorem 6.2
ex 5 proving theorem 6 220
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.

Ex. 5: Proving Theorem 6.2
ex 5 proving theorem 6 221
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

Ex. 5: Proving Theorem 6.2
ex 5 proving theorem 6 222
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.

Ex. 5: Proving Theorem 6.2
ex 5 proving theorem 6 223
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

Ex. 5: Proving Theorem 6.2
ex 5 proving theorem 6 224
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

Ex. 5: Proving Theorem 6.2
ex 5 proving theorem 6 225
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

Ex. 5: Proving Theorem 6.2
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?Ex. 6: Using parallelograms in real life
ex 6 using parallelograms in real life27
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.

Ex. 6: Using parallelograms in real life