geometry section 6-3 1112

Section 6-3Tests for Parallelograms

Wednesday, April 11, 2012

Essential Questions

How do you recognize the conditions that ensure a quadrilateral is a parallelogram?

How do you prove that a set of points forms a parallelogram in the coordinate plane?


Theorems6.9 - OPPOSITE SIDES:

6.10 - OPPOSITE ANGLES:

6.11 - DIAGONALS:

6.12 - PARALLEL CONGRUENT SET OF SIDES:


Theorems6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A

QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM

6.10 - OPPOSITE ANGLES:

6.11 - DIAGONALS:





6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A PARALLELOGRAM

6.11 - DIAGONALS:






6.11 - DIAGONALS: IF THE DIAGONALS OF A QUADRILATERAL BISECT EACH OTHER, THEN THE QUADRILATERAL IS A PARALLELOGRAM






6.11 - DIAGONALS: IF THE DIAGONALS OF A QUADRILATERAL BISECT EACH OTHER, THEN THE QUADRILATERAL IS A PARALLELOGRAM

6.12 - PARALLEL CONGRUENT SET OF SIDES: IF ONE PAIR OF OPPOSITES SIDES OF A QUADRILATERAL IS BOTH CONGRUENT AND PARALLEL, THEN THE QUADRILATERAL IS A PARALLELOGRAM


Example 1DETERMINE WHETHER THE QUADRILATERAL IS A PARALLELOGRAM.

JUSTIFY YOUR ANSWER.


Example 1DETERMINE WHETHER THE QUADRILATERAL IS A PARALLELOGRAM.

JUSTIFY YOUR ANSWER.

BOTH PAIRS OF OPPOSITE SIDES HAVE THE SAME MEASURE, SO EACH OPPOSITE PAIR IS CONGRUENT, THUS MAKING IT A

PARALLELOGRAM.


Example 2FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.



4x − 1= 3(x + 2)



4x − 1= 3(x + 2)

4x − 1= 3x + 6



4x − 1= 3(x + 2)

4x − 1= 3x + 6

x = 7



4x − 1= 3(x + 2)

4x − 1= 3x + 6

x = 7

3(y + 1) = 4y − 2



4x − 1= 3(x + 2)

4x − 1= 3x + 6

x = 7

3(y + 1) = 4y − 2

3y + 3 = 4y − 2



4x − 1= 3(x + 2)

4x − 1= 3x + 6

x = 7

3(y + 1) = 4y − 2

3y + 3 = 4y − 2

5 = y


Example 3QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO

IS A PARALLELOGRAM.



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4

= −12



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4

= −12

m(AC ) =

−3 − 12 − 3



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4

= −12

m(AC ) =

−3 − 12 − 3

=−4−1



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4

= −12

m(AC ) =

−3 − 12 − 3

=−4−1

= 4



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4

= −12

m(AC ) =

−3 − 12 − 3

=−4−1

= 4 m(TO ) =

−1− 3−2 − (−1)



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4

= −12

m(AC ) =

−3 − 12 − 3

=−4−1

= 4 m(TO ) =

−1− 3−2 − (−1)

=−4−1



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4

= −12

m(AC ) =

−3 − 12 − 3

=−4−1

= 4 m(TO ) =

−1− 3−2 − (−1)

=−4−1

= 4



IS A PARALLELOGRAM.

m(TA) =

1− 33 − (−1)

=−24

= −12

m(CO ) =−1− (−3)−2 − 2

=2−4

= −12

m(AC ) =

−3 − 12 − 3

=−4−1

= 4 m(TO ) =

−1− 3−2 − (−1)

=−4−1

= 4

SINCE EACH SET OF OPPOSITE SIDES HAVE THE SAME SLOPE, THEY ARE PARALLEL. WITH EACH SET OF OPPOSITE SIDES BEING PARALLEL, TACO IS

A PARALLELOGRAM


Example 4FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A

PARALLELOGRAM.



PARALLELOGRAM.

4x − 4 = 72



PARALLELOGRAM.

4x − 4 = 72

4x = 76



PARALLELOGRAM.

4x − 4 = 72

4x = 76

x = 19



PARALLELOGRAM.

4x − 4 = 72

4x = 76

x = 19

180− 72



PARALLELOGRAM.

4x − 4 = 72

4x = 76

x = 19

180− 72 = 108



PARALLELOGRAM.

4x − 4 = 72

4x = 76

x = 19

180− 72 = 108

8y + 8 = 108



PARALLELOGRAM.

4x − 4 = 72

4x = 76

x = 19

180− 72 = 108

8y + 8 = 108

8y = 100



PARALLELOGRAM.

4x − 4 = 72

4x = 76

x = 19

180− 72 = 108

8y + 8 = 108

8y = 100

y = 12.5


Check Your Understanding

REVIEW #1-8 ON P. 413


Problem Set


Problem Set

P. 414 #9-23 ODD, 27, 51, 53

“I AM ALWAYS DOING THAT WHICH I CAN NOT DO, IN ORDER THAT I MAY LEARN HOW TO DO IT." – PABLO PICASSO


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