1. G.9 Quadrilaterals Part 1
Parallelograms
Modified by Lisa Palen

2. Definition
A parallelogram is a quadrilateral whose opposite sides are parallel.
Its symbol is a small figure:

3. Naming a Parallelogram
A parallelogram is named using all four vertices.
You can start from any one vertex, but you must continue in a clockwise or counterclockwise direction.
For example, this can be either ABCD or ADCB.

4. Basic Properties
There are four basic properties of all parallelograms.
These properties have to do with the angles, the sides and the diagonals.

5. Opposite Sides
Theorem Opposite sides of a parallelogram are congruent.
That means that
So, if AB = 7, then _____ = 7?

6. Opposite Angles One pair of opposite angles is A and
 C. The other pair is  B and  D.

7. Opposite Angles
Theorem Opposite angles of a parallelogram are congruent.
Complete: If m  A = 75 and m  B = 105, then m  C = ______ and m  D = ______ .

8. Consecutive Angles
Each angle is consecutive to two other angles. A is consecutive with  B and  D.

9. Consecutive Angles in Parallelograms
Theorem Consecutive angles in a parallelogram are supplementary.
Therefore, m  A + m  B = 180 and m  A + m  D = 180.
If m<C = 46, then m  B = _____?
Consecutive
INTERIOR
Angles are
Supplementary!

10. Diagonals Diagonals are segments that join non-consecutive vertices.
For example, in this diagram, the only two diagonals are

11. Diagonal Property
When the diagonals of a parallelogram intersect, they meet at the midpoint of each diagonal.
So, P is the midpoint of
Therefore, they bisect each other; so and
But, the diagonals are not congruent!

12. Diagonal Property
Theorem The diagonals of a parallelogram bisect each other.

13. Parallelogram Summary
By its definition, opposite sides are parallel.
Other properties (theorems):
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
The diagonals bisect each other.

14. Examples 1. Draw HKLP. 2. Complete: HK = _______ and HP = ________ .
3. m<K = m<______ .
4. m<L + m<______ = 180.
5. If m<P = 65, then m<H = ____, m<K = ______ and m<L =______ .

15. Examples (cont’d) 6. Draw in the diagonals. They intersect at M.
7. Complete: If HM = 5, then ML = ____ .
8. If KM = 7, then KP = ____ .
9. If HL = 15, then ML = ____ .
10. If m<HPK = 36, then m<PKL = _____ .

16. Tests for Parallelograms
Part 2
Tests for Parallelograms

17. Review: Properties of Parallelograms
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
The diagonals bisect each other.

18. How can you tell if a quadrilateral is a parallelogram?
Defn: A quadrilateral is a parallelogram iff opposite sides are parallel.
Property If a quadrilateral is a parallelogram, then opposite sides are parallel.
Test If opposite sides of a quadrilateral are parallel, then it is a parallelogram.

19. Proving Quadrilaterals as Parallelograms
Theorem 1:
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram . H G
Theorem 2: E F
If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .

20. Theorem:
Theorem 3:
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. G H
then Quad. EFGH is a parallelogram. M
Theorem 4: E F
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram .
then Quad. EFGH is a parallelogram.
EM = GM and HM = FM

21. 5 ways to prove that a quadrilateral is a parallelogram.
1. Show that both pairs of opposite sides are || . [definition]
2. Show that both pairs of opposite sides are  .
3. Show that one pair of opposite sides are both || and  .
4. Show that both pairs of opposite angles are  .
5. Show that the diagonals bisect each other .

22. Examples ……
Example 1:
Find the values of x and y that ensures the quadrilateral is a parallelogram.
y+2
6x = 4x + 8
2x = 8
x = 4
2y = y + 2
y = 2 6x
4x+8 2y
Find the value of x and y that ensure the quadrilateral is a parallelogram.
Example 2:
2x + 8 = 120
2x = 112
x = 56
5y = 180
5y = 60
y = 12
(2x + 8)°
5y°
120°

23. Part 3
Rectangles
Lesson 6-3: Rectangles

24. Rectangles Definition:
A rectangle is a quadrilateral with four right angles.
Is a rectangle is a parallelogram?
Yes, since opposite angles are congruent.
Thus a rectangle has all the properties of a parallelogram.
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Lesson 6-3: Rectangles

25. Properties of Rectangles
Theorem:
If a parallelogram is a rectangle, then its diagonals are congruent.
Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles. E D C B A
Converse:
If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle.
Lesson 6-3: Rectangles

26. Properties of Rectangles
Parallelogram Properties:
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Plus:
All angles are right angles.
Diagonals are congruent.
Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles E D C B A
Lesson 6-3: Rectangles

27. Examples……. If AE = 3x +2 and BE = 29, find the value of x.
If AC = 21, then BE = _______.
If m<1 = 4x and m<4 = 2x, find the value of x.
If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.
x = 9 units
10.5 units
x = 18 units 6 5 4 3 2 1 E D C B A
m<1=50,
m<3=40,
m<4=80,
m<5=100,
m<6=40
Lesson 6-3: Rectangles

28. Lesson 6-4: Rhombus & Square
Part 4
Rhombi
and
Squares
Lesson 6-4: Rhombus & Square

29. Lesson 6-4: Rhombus & Square
Definition:
A rhombus is a quadrilateral with four congruent sides. ≡
Is a rhombus a parallelogram? ≡
Yes, since opposite sides are congruent.
Since a rhombus is a parallelogram the following are true:
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Lesson 6-4: Rhombus & Square

30. Lesson 6-4: Rhombus & Square
Note:
The four small triangles are congruent, by SSS.
This means the diagonals form four angles that are congruent, and must measure 90 degrees each. ≡ ≡
So the diagonals are perpendicular.
This also means the diagonals bisect each of the four angles of the rhombus
So the diagonals bisect opposite angles.
Lesson 6-4: Rhombus & Square

31. Properties of a Rhombus
Theorem:
The diagonals of a rhombus are perpendicular.
Theorem:
Each diagonal of a rhombus bisects a pair of opposite angles.
Note:
The small triangles are RIGHT and CONGRUENT!
Lesson 6-4: Rhombus & Square

32. Properties of a Rhombus
Since a rhombus is a parallelogram the following are true: .
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Plus:
All four sides are congruent.
Diagonals are perpendicular.
Diagonals bisect opposite angles.
Also remember: the small triangles are RIGHT and CONGRUENT! ≡ ≡
Lesson 6-4: Rhombus & Square

33. Lesson 6-4: Rhombus & Square
Rhombus Examples .....
Given: ABCD is a rhombus. Complete the following.
If AB = 9, then AD = ______.
If m<1 = 65, the m<2 = _____.
m<3 = ______.
If m<ADC = 80, the m<DAB = ______.
If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.
9 units
65°
90°
100° 10
Lesson 6-4: Rhombus & Square

34. Square
Definition:
A square is a quadrilateral with four congruent angles and four congruent sides.
Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals.
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Plus:
Four right angles.
Four congruent sides.
Diagonals are congruent.
Diagonals are perpendicular.
Diagonals bisect opposite angles.

35. Lesson 6-4: Rhombus & Square
Squares – Examples…...
Given: ABCD is a square. Complete the following.
If AB = 10, then AD = _____ and DC = _____.
If CE = 5, then DE = _____.
m<ABC = _____.
m<ACD = _____.
m<AED = _____.
10 units
10 units
5 units
90°
45°
90°
Lesson 6-4: Rhombus & Square

36. Lesson 6-5: Trapezoid & Kites
Part 5
Trapezoids
and Kites
Lesson 6-5: Trapezoid & Kites

37. Lesson 6-5: Trapezoid & Kites
Definition:
A quadrilateral with exactly one pair of parallel sides.
The parallel sides are called bases and the non-parallel sides are called legs.
Base
Trapezoid
Leg
Leg
Base
Lesson 6-5: Trapezoid & Kites

38. Lesson 6-5: Trapezoid & Kites
Median of a Trapezoid
The median of a trapezoid is the segment that joins the midpoints of the legs. (It is sometimes called a midsegment.)
Theorem - The median of a trapezoid is parallel to the bases.
Theorem - The length of the median is one-half the sum of the lengths of the bases.
Median
Lesson 6-5: Trapezoid & Kites

39. Lesson 6-5: Trapezoid & Kites
Isosceles Trapezoid
Definition:
A trapezoid with congruent legs.
Isosceles trapezoid
Lesson 6-5: Trapezoid & Kites

40. Properties of Isosceles Trapezoid
1. Both pairs of base angles of an isosceles trapezoid are congruent.
2. The diagonals of an isosceles trapezoid are congruent. B A D C
Lesson 6-5: Trapezoid & Kites

41. Lesson 6-5: Trapezoid & Kites
Definition:
A quadrilateral with two distinct pairs of congruent adjacent sides.
Diagonals of a kite are
perpendicular.
Theorem:
Lesson 6-5: Trapezoid & Kites

42. Lesson 6-5: Trapezoid & Kites
Flow Chart
Quadrilaterals
Kite
Trapezoid
Parallelogram
Rhombus
Isosceles
Trapezoid
Rectangle
Square
Lesson 6-5: Trapezoid & Kites