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?
3. Theorems
6.9 - OPPOSITE SIDES:
6.10 - OPPOSITE ANGLES:
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
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:
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:
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:
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
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.
10. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
11. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
12. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
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
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
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
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
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.
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)
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
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
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
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
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
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
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
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
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)
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
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
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
31. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
32. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
33. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
34. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
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
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
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
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
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