1. 6.3 Proving Quadrilaterals are Parallelograms
Geometry
6.3 Proving Quadrilaterals are Parallelograms

2. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Goal
Use properties of parallelograms to prove a quadrilateral is a parallelogram.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

3. Parallelogram Property
1. Opposite Sides are Parallel
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

4. Parallelogram Property
2. Opposite Sides are Congruent
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

5. Parallelogram Property
3. Opposite Angles are Congruent
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

6. Parallelogram Property
4. Consecutive Angles are Supplementary
(180 – x) x x
(180 – x)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

7. Parallelogram Property
5. Diagonals Bisect Each Other
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

8. Geometry 6.3 Proving Quadrilaterals are Parallelograms
The converse
of each of these
is also true.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

9. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Converse
If A then B.
The converse is…
If B then A.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

10. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Example
If MESA is a parallelogram then M  S and E  A. (opposite angles are congruent)
Converse:
If M  S and E  A, then MESA is a parallelogram. (because opposite angles are congruent)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

11. A quadrilateral is a parallelogram if…
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

12. Geometry 6.3 Proving Quadrilaterals are Parallelograms
To show a quadrilateral is a parallelogram only one of these is needed.
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

13. Geometry 6.3 Proving Quadrilaterals are Parallelograms
And one other way…
If one pair of sides is parallel and congruent, then the quadrilateral is a parallelogram.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

14. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Proof
2 s  by SAS
opp. sides  means it’s .
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

15. Six ways to prove a quadrilateral is a parallelogram.
opp. sides  x
180 - x
opp. sides ||
consec. s supp.
opp. s 
pair sides || and 
diags. bis.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

16. Is this a parallelogram?
110
70
YES!
70
110
Reason:
Opposite angles congruent
Or, consecutive angles supplementary.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

17. Is this a parallelogram?
10 cm
YES!
7 cm
7 cm
10 cm
Reason:
Opposite sides congruent.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

18. Is this a parallelogram?4 6
YES! 6 4
Reason:
Diagonals bisect each other.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

19. Is this a parallelogram?
NO!
7 cm
7 cm
Reason:
One pair of opp. sides ||, but not .
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

20. Is this a parallelogram?
7 cm
NO!
7 cm
Reason:
One pair of opp. sides ||, but not .
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

21. Is this a parallelogram?
9 cm
YES!
9 cm
Reason:
One pair of opp. sides || & .
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

22. Show RSTV is a parallelogram.
Since RT = RT, RST  TVR by ASA.
Using CPCTC, RS = TV and ST = RV.
RSTV is a parallelogram since opposite sides are congruent.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

23. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Now some algebra…
We will determine if a quadrilateral on the coordinate plane is a parallelogram.
Use the distance formula or the slope formula.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

24. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Coordinate Problem 1
Graph the quadrilateral with points
A(2, 4)
B(4, –3)
C(9, –6)
D(7, 1)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

25. Geometry 6.3 Proving Quadrilaterals are Parallelograms
You should have this graph: A
(2, 4) D
(7, 1) B
(4, -3) C
(9, -6)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

26. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Choices
Choose a method using the given information that will prove ABCD is a parallelogram:
Show opposite sides congruent.
Or, show they are parallel.
Or, one pair of sides congruent and parallel.
Or even the diagonals bisect each other.
But choose the easiest if possible!
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

27. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Method: Show opposite sides congruent. A B C D
Compute the length of segment AB.
(2, 4)
(7, 1)
(4, -3)
(9, -6)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

28. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Now find the length of DC. A B C D
(2, 4)
(7, 1)
AB = DC
(4, -3)
(9, -6)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

29. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Now find lengths AD & BC. A B C D
(2, 4)
(7, 1)
AD = BC
(4, -3)
(9, -6)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

30. Geometry 6.3 Proving Quadrilaterals are ParallelogramsB C D
Both pairs of opposite sides are congruent, so ABCD is a parallelogram.
(2, 4)
(7, 1)
(4, -3)
(9, -6)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

31. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Another Example
On another graph, plot the points:
R(–5, 5)
S(6, 4)
T(2, –5)
U(–9, –4)
Then connect the points to make a quadrilateral.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

32. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Your graph should be…
Find the slopes of segments RS and UT. R
(-5, 5) S
(6, 4) U
Slopes are equal, sides are parallel.
(-9, -4) T
(2, -5)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

33. Find the slopes of RU & ST.
(-5, 5) S
(6, 4) U
Slopes are equal, sides are parallel.
(-9, -4) T
(2, -5)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

34. RSTU is a parallelogram.
Reason:
Both pairs of opposite sides are parallel. R
(-5, 5) S
(6, 4) U
(-9, -4) T
(2, -5)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

35. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Try this.
A, B, and C are three vertices of a parallelogram.
Find the possible coordinates of the fourth vertex.
A(0, 1)
C(-2, -1)
B(1, -1)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

36. Geometry 6.3 Proving Quadrilaterals are Parallelograms
D could be (-3, 1)
D(-3, 1)
A(0, 1)
C(-2, -1)
B(1, -1)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

37. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Where else could D be?
A(0, 1)
D(3, 1)
C(-2, -1)
B(1, -1)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

38. Geometry 6.3 Proving Quadrilaterals are Parallelograms
Anywhere else?
A(0, 1)
C(-2, -1)
B(1, -1)
D(-1, -3)
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

39. Geometry 6.3 Proving Quadrilaterals are Parallelograms
What you’ve learned…
You can prove a quadrilateral is a parallelogram if any one of six conditions is true.
You MUST know these six methods.
You can also show a quadrilateral on the plane is a parallelogram using the distance formula or slope formula.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms

40. Six ways to prove a quadrilateral is a parallelogram.
opp. sides  x
180 - x
opp. sides ||
consec. s supp.
opp. s 
pair sides || and 
diags. bis.
November 30, 2018
Geometry 6.3 Proving Quadrilaterals are Parallelograms