2. 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?
Tuesday, April 29, 14
3. Theorems
6.9 - OPPOSITE SIDES:
6.10 - OPPOSITE ANGLES:
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
Tuesday, April 29, 14
4. Theorems
6.9 - 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:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
Tuesday, April 29, 14
5. Theorems
6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
Tuesday, April 29, 14
6. Theorems
6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A
QUADRILATERAL ARE CONGRUENT, 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:
Tuesday, April 29, 14
7. Theorems
6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A
QUADRILATERAL ARE CONGRUENT, 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
Tuesday, April 29, 14
9. Example 1
DETERMINE 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.
Tuesday, April 29, 14
10. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
Tuesday, April 29, 14
11. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
Tuesday, April 29, 14
12. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
Tuesday, April 29, 14
13. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
Tuesday, April 29, 14
14. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
Tuesday, April 29, 14
15. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
3y + 3 = 4y − 2
Tuesday, April 29, 14
16. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
3y + 3 = 4y − 2
5 = y
Tuesday, April 29, 14
17. Example 3
QUADRILATERAL 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.
Tuesday, April 29, 14
18. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
Tuesday, April 29, 14
19. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
Tuesday, April 29, 14
20. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
Tuesday, April 29, 14
21. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
Tuesday, April 29, 14
22. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
Tuesday, April 29, 14
23. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
= −
1
2
Tuesday, April 29, 14
24. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
= −
1
2
m(AC) =
−3 − 1
2 − 3
Tuesday, April 29, 14
25. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
= −
1
2
m(AC) =
−3 − 1
2 − 3
=
−4
−1
Tuesday, April 29, 14
26. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
= −
1
2
m(AC) =
−3 − 1
2 − 3
=
−4
−1
= 4
Tuesday, April 29, 14
27. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
= −
1
2
m(AC) =
−3 − 1
2 − 3
=
−4
−1
= 4 m(TO) =
−1− 3
−2 − (−1)
Tuesday, April 29, 14
28. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
= −
1
2
m(AC) =
−3 − 1
2 − 3
=
−4
−1
= 4 m(TO) =
−1− 3
−2 − (−1)
=
−4
−1
Tuesday, April 29, 14
29. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
= −
1
2
m(AC) =
−3 − 1
2 − 3
=
−4
−1
= 4 m(TO) =
−1− 3
−2 − (−1)
=
−4
−1
= 4
Tuesday, April 29, 14
30. Example 3
QUADRILATERAL 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.
m(TA) =
1− 3
3 − (−1)
=
−2
4
= −
1
2
m(CO) =
−1− (−3)
−2 − 2
=
2
−4
= −
1
2
m(AC) =
−3 − 1
2 − 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
Tuesday, April 29, 14
31. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
Tuesday, April 29, 14
32. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
Tuesday, April 29, 14
33. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
Tuesday, April 29, 14
34. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
Tuesday, April 29, 14
35. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72
Tuesday, April 29, 14
36. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72 = 108
Tuesday, April 29, 14
37. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72 = 108
8y + 8 = 108
Tuesday, April 29, 14
38. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72 = 108
8y + 8 = 108
8y = 100
Tuesday, April 29, 14
39. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72 = 108
8y + 8 = 108
8y = 100
y = 12.5
Tuesday, April 29, 14
41. Problem Set
P. 413 #1-23 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
Tuesday, April 29, 14
