Proving Quadrilaterals are //ograms.

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Proving Quadrilaterals are //ograms.

• Chapter 6.3

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Proving Quadrilaterals are //ograms.Methods 1-3

1. If opposite sides of a quadrilateral are //, then
   it is a //ogram. (definition)
2. If both pairs of opposite sides of a
   quadrilateral are ?, then it is a //ogram.
3. If both pairs of opposite angles are ?, then it
   is a //ogram.

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Proving Quadrilaterals are //ograms.Methods 4-6

• If an angle of a quadrilateral is supplementary
  to both of its consecutive angles, then it is a
  //ogram.
• If the diagonals bisect each other, then it is a
  parallelogram.
• If one pair of opposite sides of a quadrilateral
  are // and ?, then it is a parallelogram. (new)

Additional Test for a //ogram
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Is this a Parallelogram? Why?
Yes. Opposite Angles are Congruent.
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Is this a Parallelogram? Why?
No, not enough information.
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Is this a Parallelogram? Why?
Yes. Opposite Sides are Parallel (definition).
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Is this a Parallelogram? Why?
Yes. One pair of opposite sides are parallel and
congruent.
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Is this a Parallelogram? Why?
120o
60o
120o
Yes. An angle is supplementary to both of its
consecutive angles.
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Is this a Parallelogram? Why?
No, not enough information.
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Is this a Parallelogram? Why?
Yes. Opposite Sides are Congruent.
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Is this a Parallelogram? Why?
120o
60o
No, not enough information.
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Is this a Parallelogram? Why?
Yes. Diagonals bisect each other.
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Is this a Parallelogram? Why?
No, not enough information.
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Is this a Parallelogram? Why?
A
B
?ABC ? ?CDA
D
C
Yes, Opposite sides are congruent. Others can be
proven as well.
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Reminders Coordinate Proofs

• Distance Formula

• Midpoint Formula

• Slope?// lines have equal slope

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Prove this is a Parallelogram
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Prove this is a Parallelogram

• Slope Method
• Prove AB//CD and BC//AD
• Use slope formula and show that their slopes are
  equal.

• Distance Method
• Prove AB CD and BC AD
• Use Distance Formula to show that their lengths
  are equal.

• Slope Distance
• Prove AB CD and AB // CD
• Use Distance Formula to show that their lengths
  are equal and use slope formula to show that
  their slopes are equal.

• Midpoint Method
• Prove the diagonals bisect each other
• Show that the diagonals have the same midpoint.

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Proof Since ?XVY ? ?ZVW and ?XVW ? ?ZVY,

by
CPCTC. By which method would you prove WXYZ is a
parallelogram?
A. Both pairs of opp. sides ?. B. Both pairs of
opp. ?s ?. C. One pair of opp. sides both ? and
. D. Diagonals bisect each other
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Properties of Parallelograms
Determine whether the quadrilateral is a
parallelogram. Justify your answer.
Answer Each pair of opposite sides has the same
measure. Therefore, they are congruent. If both
pairs of opposite sides of a quadrilateral are
congruent, the quadrilateral is a parallelogram.
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Which method would prove the quadrilateral is a
parallelogram?
A. Both pairs of opp. sides . B. Both pairs of
opp. sides ?. C. Both pairs of opp. ?s ?. D. One
pair of opp. sides both and ?.
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Find x so that the quadrilateral is a //ogram.
Opposite sides of a //ogram are congruent.
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Find m so that the quadrilateral is a //ogram.
A. m 2 B. m 3 C. m 6 D. m 8
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Use Slope and Distance
COORDINATE GEOMETRY Determine whether the figure
with vertices A(3, 0), B(1, 3), C(3, 2), and
D(1, 1) is a parallelogram. Use the Slope
Formula.
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slopes ? // Lines
Opp. Sides are // ? //ogram
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A. yes B. no C. cannot be determined

1. A
2. B
3. C

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Homework

• Chapter 6.3
• Pg 337
• 3-14, 20-25, 26, 28, 45-48