1. Lesson 6-1: Parallelogram
Parallelograms
Lesson 6-1: Parallelogram

2. Lesson 6-1: Parallelogram
Definition:
A quadrilateral whose opposite sides are parallel. C B A D
Symbol:
a smaller version
of a parallelogram
Naming:
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, the figure above can be either ABCD or ADCB.
Lesson 6-1: Parallelogram

3. Properties of ParallelogramB
Properties of Parallelogram P D C
1. Both pairs of opposite sides are parallel
2. Both pairs of opposite sides are congruent.
3. Both pairs of opposite angles are congruent.
4. Consecutive angles are supplementary.
5. Diagonals bisect each other but are not congruent
P is the midpoint of
Lesson 6-1: Parallelogram

4. Lesson 6-1: ParallelogramH K
Examples M P L
Draw HKLP.
HK = _______ and HP = ________ .
m<K = m<______ .
m<L + m<______ = 180.
If m<P = 65, then m<H = ____,m<K = ______ and m<L =____.
Draw the diagonals with their point of intersection labeled M.
If HM = 5, then ML = ____
If KM = 7, then KP = ____
If HL = 15, then ML = ____
If m<HPK = 36, then m<PKL = _____ . PL KL P
P or K
115°
115°
65
5 units
14 units
7.5 units
36;
(Alternate interior angles are congruent.)
Lesson 6-1: Parallelogram

5. 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 .

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

7. Rectangles Definition:
A rectangle is a parallelogram with four right angles.
A rectangle is a special type of parallelogram.
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

8. 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

9. 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

10. Lesson 6-4: Rhombus & Square
Definition:
A rhombus is a parallelogram with four congruent sides. ≡ ≡
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

11. Properties of a Rhombus
Theorem:
The diagonals of a rhombus are perpendicular.
Theorem:
Each diagonal of a rhombus bisects a pair of opposite angles.
Lesson 6-4: Rhombus & Square

12. 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

13. Lesson 6-4: Rhombus & Square
Definition:
A square is a parallelogram 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.
Four right angles.
Four congruent sides.
Consecutive angles are supplementary.
Diagonals are congruent.
Diagonals bisect each other.
Diagonals are perpendicular.
Each diagonal bisects a pair of opposite angles.
Lesson 6-4: Rhombus & Square

14. 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

15. 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.
Trapezoid
Base
Leg
An Isosceles trapezoid is a trapezoid with congruent legs.
Isosceles trapezoid
Lesson 6-5: Trapezoid & Kites

16. 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
Base Angles D C
Lesson 6-5: Trapezoid & Kites

17. Lesson 6-5: Trapezoid & Kites
Median of a Trapezoid
The median of a trapezoid is the segment that joins the midpoints of the legs.
The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the measures of the bases.
Median
Lesson 6-5: Trapezoid & Kites

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