Section 6.3: Proving Quadrilaterals are Parallelogram

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Section 6.3 Proving Quadrilaterals are
Parallelogram
Goals

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

Anchor

• Identify and use properties of quadrilaterals
• Relate slope to parallelism
• Calculate the distance between two points

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How do we prove that a quadrilateral is a
parallelogram?
Theorem 6.6
Theorem 6.7
Theorem 6.8
Theorem 6.9
Theorem 6.10
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Proving Quadrilaterals are Parallelograms

• Show that both pairs of opp. sides are
• Show that both pairs of opp. sides are ?
• Show that both pairs of opp. angles are ?
• Show that one angle is supp. to both consecutive
  angles
• Show that the diagonals bisect each other
• Show that one pair of sides are ? and

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Show that quad ABCD is a parallelogram.
A ( 2 , -1 ) , B ( 1 , 3 ) , C ( 6 , 5 ) , D ( 7
, 1 )
Method 1
Slope of AB Slope of BC Slope of CD Slope
of AD
If both pairs of opposite sides of a
quadrilateral are parallel, then the
quadrilateral is a parallelogram
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Show that quad ABCD is a parallelogram.
A ( 2 , -1 ) , B ( 1 , 3 ) , C ( 6 , 5 ) , D ( 7
, 1 )
Method 2
Length of AB Length of BC Length of CD
Length of AD
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram
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Show that quad ABCD is a parallelogram.
A ( 2 , -1 ) , B ( 1 , 3 ) , C ( 6 , 5 ) , D ( 7
, 1 )
Method 3
Slope of AB Slope of CD Length of AB
Length of CD
If one pair of opposite sides of a quadrilateral
are congruent and parallel, then the
quadrilateral is a parallelogram
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Show that quad QRST is a parallelogram.
Q ( -6 , 2 ) , R ( 3 , 11 ) , S ( 10 , 9 ) , T (
1 , 0 )
Try all three of the methods.
Slope of QR Slope of RS Slope of ST Slope
of QT
Length of QR Length of RS Length of ST
Length of QT
Slope of QR Slope of ST Length of QR
Length of ST
Identify any quadrilateral that is a
parallelogram. Show all work to support you
answer.
1) G ( -3 , 1 ) H ( 4 , 1 ) I ( 3 , 6 ) , J (
-1 , 6 )
2) K ( -2 , 2 ) , L ( 1 , 1 ) , M ( 4 , 4 ) , N
( 1 , 4 )
3) W ( 3 , -1 ) , X ( 4 , 2 ) , Y ( 1 , 5 ) , Z
( 0 , 2 )