Properties of a parallelograms

1. Lesson:
Objectives:
6.3 Tests for Parallelograms
 To Identify the 5 CONDITIONS that GUARANTEE
that a QUADRILATERAL is a PARALLELOGRAM
 To Use the 5 CONDITIONS to SOLVE Problems

2. GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:

3. GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL

4. GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
2. OPPOSITE SIDES are CONGRUENT.

5. GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
2. OPPOSITE SIDES are CONGRUENT.
3. OPPOSITE ANGLES are CONGRUENT.

6. GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
2. OPPOSITE SIDES are CONGRUENT.
3. OPPOSITE ANGLES are CONGRUENT.
4. CONSECUTIVE ANGLES are SUPPLEMENTARY.

7. GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
2. OPPOSITE SIDES are CONGRUENT.
3. OPPOSITE ANGLES are CONGRUENT.
4. CONSECUTIVE ANGLES are SUPPLEMENTARY.
5. DIAGONALS Bisect each other.

8. GEOMETRY 6.3
Which, if any, of the Properties of a Parallelogram
PROVE that a Quadrilateral IS a Parallelogram?

9. GEOMETRY 6.3
IF a QUADRILATERAL has OPPOSITE SIDES
that are PARALLEL
Is it a PARALLELOGRAM?

10. GEOMETRY 6.3
IF a QUADRILATERAL has OPPOSITE SIDES
that are PARALLEL
Is it a PARALLELOGRAM?
YES – the DEFINITION of a PARALLELOGRAM
is a Quadrilateral for which
OPPOSITE SIDES are Parallel!

11. GEOMETRY 6.3
IF a QUADRILATERAL has
ONE PAIR of OPPOSITE SIDES that are CONGRUENT,
Is it a PARALLELOGRAM?

12. GEOMETRY 6.3
IF a QUADRILATERAL has
ONE PAIR of OPPOSITE SIDES that are CONGRUENT,
Is it a PARALLELOGRAM?
Can you DRAW a COUNTEREXAMPLE?

13. GEOMETRY 6.3
IF a QUADRILATERAL has
BOTH PAIR of OPPOSITE SIDES that are CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?

14. GEOMETRY 6.3
IF a QUADRILATERAL has
BOTH PAIR of OPPOSITE SIDES that are CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?

16. GEOMETRY 6.3
IF a QUADRILATERAL has
ONE PAIR of OPPOSITE ANGLES that are CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?

17. GEOMETRY 6.3
IF a QUADRILATERAL has
BOTH PAIRs of OPPOSITE ANGLES that are CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?

18. GEOMETRY 6.3
GEOMETRY 6.3
Given: Angles T and R are Congruent
Angles Q and S are Congruent
Prove: QRST is a Parallelogram

19. GEOMETRY 6.3
IF a QUADRILATERAL has
DIAGONALS that Bisect each other,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?

20. GEOMETRY 6.3

21. GEOMETRY 6.3
IF a QUADRILATERAL has
ONE PAIR of OPPOSITE SIDES that is BOTH
PARALLEL and CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?

22. GEOMETRY 6.3

23. GEOMETRY 6.3

24. GEOMETRY 6.3

25. GEOMETRY 6.3

26. GEOMETRY 6.3

27. GEOMETRY 6.3

28. GEOMETRY 6.3

29. GEOMETRY 6.3

30. GEOMETRY 6.3

31. COORDINATE GEOMETRY Determine whether the
figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and
D(1, –1) is a parallelogram.
Three Methods:
1. SLOPE formula
2. DISTANCE formula
3. MIDPOINT formula

32. Geometry 6.3
You should be able to:
 Determine is a Quadrilateral is a PARALLEOGRAM
 Determine if a CONDITION defines a PARALLELOGRAM

33. Lesson:
Objectives:
6.4 Rectangles
 To Identify the PROPERTIES of RECTANGLES
 To Use the Rectangle Properties to SOLVE Problems
 To Identify the PROPERTIES of SQUARES and RHOMBI
 To use the Squares and Rhombi Properties to SOLVE
Problems

34. GEOMETRY 6.4
A RECTANGLE is:

35. GEOMETRY 6.4
GEOMETRY 6.4
A RECTANGLE is:
A QUADRILATERAL

36. GEOMETRY 6.4
GEOMETRY 6.4
A RECTANGLE is:
A QUADRILATERAL
A PARALLELOGRAM

37. GEOMETRY 6.4
GEOMETRY 6.4
A RECTANGLE is:
A QUADRILATERAL
A PARALLELOGRAM
with 4 Right Angles

38. GEOMETRY 6.4
PROPERTIES of a Rectangle:
 Same as a Parallelogram

39. GEOMETRY 6.4
PROPERTIES of a Rectangle:
 Same as a Parallelogram
 Opposite Sides are Parallel
 Opposite Sides are Congruent
 Opposite Angles are Congruent
 Consecutive Sides are Supplementary
 Diagonals BISECT each other.

40. GEOMETRY 6.4
PROPERTIES of a Rectangle:
 Same as a Parallelogram
 Opposite Sides are Parallel
 Opposite Sides are Congruent
 Opposite Angles are Congruent
 Consecutive Sides are Supplementary
 Diagonals BISECT each other.
 All ANGLES are CONGRUENT

41. GEOMETRY 6.4
PROPERTIES of a Rectangle:
 Same as a Parallelogram
 Opposite Sides are Parallel
 Opposite Sides are Congruent
 Opposite Angles are Congruent
 Consecutive Sides are Supplementary
 Diagonals BISECT each other.
 All ANGLES are CONGRUENT
 DIAGONALS are

42. GEOMETRY 6.4
PROOF
 DIAGONALS are
A B
C
D
GIVEN:
PROVE:
ABCD is a RECTANGLE
AC and BD are diagonals
AC BD


43. GEOMETRY 6.4
Find X
M N
O
P
C
MO x
NP
 

2 8
23

44. GEOMETRY 6.4
GEOMETRY 6.4
Find X
GEOMETRY 6.4
M N
O
P
C
CN x
CO x
 
 
2
1
3 11

45. GEOMETRY 6.4
GEOMETRY 6.4
Find X
GEOMETRY 6.4
GEOMETRY 6.4
Find X
GEOMETRY 6.4
M N
O
P
C
MO x
PC x
 
 
4 13
7

46. GEOMETRY 6.4
GEOMETRY 6.4
TRUE or FALSE?
If a QUADRILATERAL has OPPOSITE SIDES
that are CONGRUENT,
then it is a RECTANGLE.

47. GEOMETRY 6.4
K L
M
N
10
1
2
3
4
5
6
7
8
9
C
m
Find m
m
m
 
 
 
 
1 70
2
5
6
:

48. GEOMETRY 6.4
K L
M
N
10
1
2
3
4
5
6
7
8
9
C
GEOMETRY 6.4
m
Find m
m
m
 
 
 
 
9 128
6
7
8
:

49. GEOMETRY 6.4
K L
M
N
10
1
2
3
4
5
6
7
8
9
C
GEOMETRY 6.4
m
Find m
m
 
 
 
5 36
2
3

50. Kyle is building a barn for his horse. He measures the
diagonals of the door opening to make sure that they bisect
each other and they are congruent. How does he know that
the measure of each corner is 90?

51. Quadrilateral ABCD has vertices A(–2, 1), B(4, 3),
C(5, 0), and D(–1, –2). Determine whether ABCD is a
rectangle.
Methods:
1. Slope
2. Distance