The document discusses properties of parallelograms and provides examples of determining if a quadrilateral is a parallelogram. It defines four theorems for identifying parallelograms based on opposite sides, opposite angles, bisecting diagonals, and parallel/congruent sides. Examples solve systems of equations to find values of variables such that the quadrilaterals satisfy parallelogram properties. One example uses slopes of side segments to show a quadrilateral is a parallelogram due to parallel opposite sides.

1. Section 6-3
Tests for Parallelograms

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

8. Example 1
DETERMINE WHETHER THE QUADRILATERAL IS A PARALLELOGRAM.
JUSTIFY YOUR ANSWER.

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