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:
3-14, 20-25, 3-14, 20-25, 26, 28, 45- 26, 28, 45-4848