1. Review In ABC, centroid D is on median AM. AD = x + 6 DM = 2x – 12
Find AM.
Did you draw a picture?
Did you think about the key word?

2. Parallelogram
A quadrilateral with both pairs of opposite sides parallel.

3. Parallelograms have Properties
Click to view 3

4. Properties of Parallelograms Toolkit 6.2
Today’s Goal(s):
To use relationships among sides and among angles of parallelograms.
To use relationships involving diagonals of parallelograms or transversals.

5. Properties of Every Parallelogram:
Both pairs of opposite sides are congruent.
Both pairs of opposite angles are congruent.
Consecutive adjacent angles are supplementary.
Diagonals bisect each other.

6. ANGLES… Opposite vs. Consecutive
CONGRUENT
SUPPLEMENTARY

7. 5 Properties of a Parallelogram…
Opposite sides are congruent.
Opposite sides are also parallel.
Opposite angles are congruent.
The diagonals bisect each other.
Consecutive angles are supplementary.
Mark the diagrams!

8. 6.3 Examples Determine whether the quadrilateral must be a parallelogram. Explain.

9. #1 Find the value of x in each parallelogram.
x = 60
a = 18

10. #2 Find the measures of the numbered angles for each parallelogram.
m1 = 38 m1 = 81 m1 = 95
m2 = 32 m2 = 28 m2 = 37
m3 = 110 m3 = 71 m3 = 37

11. #3 Find the value of x for which ABCD must be a parallelogram.
x = 5
x = 5

12. #4 Use the given information to find the lengths of all four sides of  ABCD.
The perimeter is 66 cm.
AD is 5 cm less than three times AB.
x = 9.5
BC = AD = 23.5
AB = CD = 9.5

13. #5 In a parallelogram one angle is 9 times the size of another
#5 In a parallelogram one angle is 9 times the size of another. Find the measures of the angles.
18 and 162

15. “Special Parallelograms”
Rectangle
Rhombus
Square

16. Rectangle
A parallelogram with four right angles.

17. What are the properties of a rectangle?
All the properties of every parallelogram. (What are these properties?)
All four angles are right angles.
The diagonals are congruent.

18. Rectangle A rectangle has ALL the properties of a parallelogram, PLUS…
All four angles of a rectangle are 90.
The diagonals of a rectangle are congruent.
AC  BD

19. Ex.2: Find the length of the diagonals of rectangle ABCD.
a.) AC = 2y + 4 and BD = 6y – 5
b.) AC = 5y – 9 and BD = y + 5

20. Rhombus
A parallelogram with four congruent sides.

21. What are the properties of a rhombus?
All the properties of every parallelogram.
The diagonals are perpendicular.
Each diagonal bisects two angles of the rhombus.

22. Rhombus A rhombus has ALL the properties of a parallelogram, PLUS…
All four sides of a rhombus are congruent.
Each diagonal of a rhombus BISECTS two angles.
The diagonals of a rhombus are perpendicular.

23. Ex.1: Find the measures of the numbered angles in each rhombus.
a b.

24. Square
A parallelogram with four congruent sides and four right angles.

25. Square
A square has ALL the properties of a parallelogram, PLUS ALL the properties of a rhombus, PLUS ALL the properties of a rectangle.
A square is a parallelogram, a rectangle, and a rhombus!

26. So, that means that in a square…
All four sides are congruent.
All four angles are 90.
The diagonals BISECT each other.
The diagonals are perpendicular.
The diagonals are congruent.

27. Summary Slide What is a parallelogram?
Properties of Every Parallelogram:
What is a rectangle?
What are the properties of a rectangle?
What is a rhombus?
What are the properties of a rhombus?
What is a square?
What are the properties of a square?

28. Your turn: Do: On your paper, list the properties of a square.
Think: How can you use these properties to determine the measures of sides, angles, and diagonals of a parallelogram? Be ready to share your thoughts!
Home

29. Kite
A quadrilateral with 2 pairs of adjacent sides congruent and NO opposite sides congruent.

30. Trapezoid
A quadrilateral with exactly one pair of parallel sides.

31. Isosceles Trapezoid
A trapezoid whose nonparallel opposite sides are CONGRUENT.

33. If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

34. Do you remember…? 5 Properties of a Parallelogram
Hint: 2-sides, 2-angles, 1-diagonals

35. Proving a shape is a Parallelogram Toolkit 6.3
Today’s Goal(s):
To use relationships among sides and among angles to determine whether a shape is a parallelogram.

36. There are 5 ways to PROVE that a shape is a parallelogram:
Show that BOTH pairs of opposite SIDES are parallel.
Show that BOTH pairs of opposite sides are congruent.
Show that BOTH pairs of opposite ANGLES are congruent.
Show that the DIAGONALS bisect each other.
Show that ONE PAIR of OPPOSITE sides is both congruent & parallel.

37. Let’s set up some proofs! 

38. You try this one…

39. Ex.2: Two-Column Proof

40. Hmm… is there more than one way to write this proof?
Statements Reasons

41. Special Parallelograms Toolkit #6.4
Today’s Goal(s):
To use properties of diagonals of rhombuses and rectangles.

42. EOC Review #6 Wednesday
ABC has a perimeter of 10x. The midpoints of the triangle are joined together to form another triangle. What is the difference in the perimeters of the two triangles?
Where is the center of the largest circle that you could draw INSIDE a given triangle?

43. EOC Review #6 Tuesday
Plot the following points on a graph and decide if AD is an altitude, median, angle bisector or perpendicular bisector.
A(6,7) B(8,2) C(2,2) D(6,2)
Point C is a centroid.
Solve for x.