6.3 Proving Quadrilaterals are Parallelograms

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6.3 Proving Quadrilaterals are Parallelograms

• Geometry
• Mrs. Spitz
• Spring 2005

2
Objectives

• Prove that a quadrilateral is a parallelogram.
• Use coordinate geometry with parallelograms.

3
Assignment

• pp. 342-343 1-19, 25, 26, 29

4
Theorems

• Theorem 6.6 If both pairs of opposite sides of
  a quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

ABCD is a parallelogram.
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Theorems

• Theorem 6.7 If both pairs of opposite angles of
  a quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

ABCD is a parallelogram.
6
Theorems

• Theorem 6.8 If an angle of a quadrilateral is
  supplementary to both of its consecutive angles,
  then the quadrilateral is a parallelogram.

(180 x)
x
x
ABCD is a parallelogram.
7
Theorems

• Theorem 6.9 If the diagonals of a quadrilateral
  bisect each other, then the quadrilateral is a
  parallelogram.

ABCD is a parallelogram.
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Ex. 1 Proof of Theorem 6.6

• Statements
• AB ? CD, AD ? CB.
• AC ? AC
• ?ABC ? ?CDA
• ?BAC ? ?DCA, ?DAC ? ?BCA
• ABCD, AD CB.
• ABCD is a ?

• Reasons
• Given

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Ex. 1 Proof of Theorem 6.6

• Statements
• AB ? CD, AD ? CB.
• AC ? AC
• ?ABC ? ?CDA
• ?BAC ? ?DCA, ?DAC ? ?BCA
• ABCD, AD CB.
• ABCD is a ?

• Reasons
• Given
• Reflexive Prop. of Congruence

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Ex. 1 Proof of Theorem 6.6

• Statements
• AB ? CD, AD ? CB.
• AC ? AC
• ?ABC ? ?CDA
• ?BAC ? ?DCA, ?DAC ? ?BCA
• ABCD, AD CB.
• ABCD is a ?

• Reasons
• Given
• Reflexive Prop. of Congruence
• SSS Congruence Postulate

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Ex. 1 Proof of Theorem 6.6

• Statements
• AB ? CD, AD ? CB.
• AC ? AC
• ?ABC ? ?CDA
• ?BAC ? ?DCA, ?DAC ? ?BCA
• ABCD, AD CB.
• ABCD is a ?

• Reasons
• Given
• Reflexive Prop. of Congruence
• SSS Congruence Postulate
• CPCTC

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Ex. 1 Proof of Theorem 6.6

• Statements
• AB ? CD, AD ? CB.
• AC ? AC
• ?ABC ? ?CDA
• ?BAC ? ?DCA, ?DAC ? ?BCA
• ABCD, AD CB.
• ABCD is a ?

• Reasons
• Given
• Reflexive Prop. of Congruence
• SSS Congruence Postulate
• CPCTC
• Alternate Interior ?s Converse

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Ex. 1 Proof of Theorem 6.6

• Statements
• AB ? CD, AD ? CB.
• AC ? AC
• ?ABC ? ?CDA
• ?BAC ? ?DCA, ?DAC ? ?BCA
• ABCD, AD CB.
• ABCD is a ?

• Reasons
• Given
• Reflexive Prop. of Congruence
• SSS Congruence Postulate
• CPCTC
• Alternate Interior ?s Converse
• Def. of a parallelogram.

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Ex. 2 Proving Quadrilaterals are Parallelograms

• As the sewing box below is opened, the trays are
  always parallel to each other. Why?

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Ex. 2 Proving Quadrilaterals are Parallelograms

• Each pair of hinges are opposite sides of a
  quadrilateral. The 2.75 inch sides of the
  quadrilateral are opposite and congruent. The 2
  inch sides are also opposite and congruent.
  Because opposite sides of the quadrilateral are
  congruent, it is a parallelogram. By the
  definition of a parallelogram, opposite sides are
  parallel, so the trays of the sewing box are
  always parallel.

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Another Theorem

• Theorem 6.10If one pair of opposite sides of a
  quadrilateral are congruent and parallel, then
  the quadrilateral is a parallelogram.
• ABCD is a
• parallelogram.

B
C
A
D
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Ex. 3 Proof of Theorem 6.10Given BCDA, BC ?
DAProve ABCD is a ?

• Statements
• BC DA
• ?DAC ? ?BCA
• AC ? AC
• BC ? DA
• ?BAC ? ?DCA
• AB ? CD
• ABCD is a ?

• Reasons
• 1. Given

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Ex. 3 Proof of Theorem 6.10Given BCDA, BC ?
DAProve ABCD is a ?

• Statements
• BC DA
• ?DAC ? ?BCA
• AC ? AC
• BC ? DA
• ?BAC ? ?DCA
• AB ? CD
• ABCD is a ?

• Reasons
• Given
• Alt. Int. ?s Thm.

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Ex. 3 Proof of Theorem 6.10Given BCDA, BC ?
DAProve ABCD is a ?

• Statements
• BC DA
• ?DAC ? ?BCA
• AC ? AC
• BC ? DA
• ?BAC ? ?DCA
• AB ? CD
• ABCD is a ?

• Reasons
• Given
• Alt. Int. ?s Thm.
• Reflexive Property

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Ex. 3 Proof of Theorem 6.10Given BCDA, BC ?
DAProve ABCD is a ?

• Statements
• BC DA
• ?DAC ? ?BCA
• AC ? AC
• BC ? DA
• ?BAC ? ?DCA
• AB ? CD
• ABCD is a ?

• Reasons
• Given
• Alt. Int. ?s Thm.
• Reflexive Property
• Given

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Ex. 3 Proof of Theorem 6.10Given BCDA, BC ?
DAProve ABCD is a ?

• Statements
• BC DA
• ?DAC ? ?BCA
• AC ? AC
• BC ? DA
• ?BAC ? ?DCA
• AB ? CD
• ABCD is a ?

• Reasons
• Given
• Alt. Int. ?s Thm.
• Reflexive Property
• Given
• SAS Congruence Post.

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Ex. 3 Proof of Theorem 6.10Given BCDA, BC ?
DAProve ABCD is a ?

• Statements
• BC DA
• ?DAC ? ?BCA
• AC ? AC
• BC ? DA
• ?BAC ? ?DCA
• AB ? CD
• ABCD is a ?

• Reasons
• Given
• Alt. Int. ?s Thm.
• Reflexive Property
• Given
• SAS Congruence Post.
• CPCTC

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Ex. 3 Proof of Theorem 6.10Given BCDA, BC ?
DAProve ABCD is a ?

• Statements
• BC DA
• ?DAC ? ?BCA
• AC ? AC
• BC ? DA
• ?BAC ? ?DCA
• AB ? CD
• ABCD is a ?

• Reasons
• Given
• Alt. Int. ?s Thm.
• Reflexive Property
• Given
• SAS Congruence Post.
• CPCTC
• If opp. sides of a quad. are ?, then it is a ?.

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Objective 2 Using Coordinate Geometry

• When a figure is in the coordinate plane, you can
  use the Distance Formula (seeit never goes away)
  to prove that sides are congruent and you can use
  the slope formula (see how you use this again?)
  to prove sides are parallel.

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Ex. 4 Using properties of parallelograms

• Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1)
  are the vertices of a parallelogram.

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Ex. 4 Using properties of parallelograms

• Method 1Show that opposite sides have the same
  slope, so they are parallel.
• Slope of AB.
• 3-(-1) - 4
• 1 - 2
• Slope of CD.
• 1 5 - 4
• 7 6
• Slope of BC.
• 5 3 2
• 6 - 1 5
• Slope of DA.
• - 1 1 2
• 2 - 7 5
• AB and CD have the same slope, so they are
  parallel. Similarly, BC DA.

Because opposite sides are parallel, ABCD is a
parallelogram.
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Ex. 4 Using properties of parallelograms

• Method 2Show that opposite sides have the same
  length.
• ABv(1 2)2 3 (- 1)2 v17
• CDv(7 6)2 (1 - 5)2 v17
• BCv(6 1)2 (5 - 3)2 v29
• DA v(2 7)2 (-1 - 1)2 v29
• AB ? CD and BC ? DA. Because both pairs of
  opposites sides are congruent, ABCD is a
  parallelogram.

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Ex. 4 Using properties of parallelograms

• Method 3Show that one pair of opposite sides is
  congruent and parallel.
• Slope of AB Slope of CD -4
• ABCD v17
• AB and CD are congruent and parallel, so ABCD is
  a parallelogram.

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Reminder

• Quiz after this section
• Make sure your definitions and postulates/definiti
  ons have been completed. After this section, I
  will not give you credit if it is late.