Chapter 6.3Chapter 6.3

1.1. If opposite sides of a quadrilateral If opposite sides of a quadrilateral are //, then it is a //ogram. are //, then it is a //ogram. (definition)(definition)

2.2. If both pairs of opposite sides of a If both pairs of opposite sides of a quadrilateral are quadrilateral are , then it is a , then it is a //ogram.//ogram.

3.3. If both pairs of opposite angles are If both pairs of opposite angles are , then it is a //ogram., then it is a //ogram.

4.4. If an angle of a quadrilateral is If an angle of a quadrilateral is supplementary to supplementary to bothboth of its consecutive of its consecutive angles, then it is a //ogram.angles, then it is a //ogram.

5.5. If the diagonals bisect each other, then it is a If the diagonals bisect each other, then it is a parallelogram.parallelogram.

6.6. If one pair of opposite sides of a quadrilateral If one pair of opposite sides of a quadrilateral are are //// and and , then it is a parallelogram. (new), then it is a parallelogram. (new)

Additional Test for a //ogram

Yes. Opposite Angles are Congruent.

No, not enough information.

Yes. Opposite Sides are Parallel (definition).

Yes. One pair of opposite sides are parallel and congruent.

Yes. An angle is supplementary to both of its consecutive angles.

60o 120o

120o

No, not enough information.

Yes. Opposite Sides are Congruent.

60o

120o

No, not enough information.

Yes. Diagonals bisect each other.

No, not enough information.

Yes, Opposite sides are congruent.

Others can be proven as well.

A

D C

B

ABC CDA

Distance FormulaDistance Formula

212

212 )()( yyxxd

• Midpoint Formula

2

,2

),( 2121 yyxxyx mm

• Slope// lines have equal slope

12

12

xx

yyslope

Slope MethodSlope Method Prove AB//CD Prove AB//CD

and BC//ADand BC//AD Use slope Use slope

formula and formula and show that show that their slopes their slopes are equal.are equal.

Distance Distance MethodMethod

Prove AB = CD Prove AB = CD and BC = ADand BC = AD

Use Distance Use Distance Formula to Formula to show that their show that their lengths are lengths are equal.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.

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

Proof: Since ΔXVY ΔZVW and ΔXVW ΔZVY, by CPCTC. By which method would you prove WXYZ is a parallelogram?

Properties of ParallelogramsDetermine 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.

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 .

Which method would prove the quadrilateral is a parallelogram?

Find x so that the quadrilateral is a //ogram.

Opposite sides of a //ogram are congruent.

A. m = 2

B. m = 3

C. m = 6

D. m = 8

Find m so that the quadrilateral is a //ogram.

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.

Use Slope and Distance

2

3AB of slope

2

3CD of slope

4

1BC of slope

4

1AD of slope

= slopes // Lines Opp. Sides are // //ogram

1. A2. B3. C

Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.

A(–1, –2), B(–3, 1), C(1, 2), D(3, –1); Slope Formula

A. yesB. noC. cannot be

determined

Chapter 6.3Chapter 6.3 Pg 337:Pg 337:

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