1. QUADRILATERALS 4-SIDED POLYGONS

2. Properties of Parallelograms Review
By its definition, opposite sides are parallel.
Other properties :
Opposite sides are congruent.
The diagonals bisect each other.
Opposite angles are congruent.
Consecutive angles are supplementary.

3. Rectangles

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

5. 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.

6. 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=50O,
m<3=40O,
m<4=80O,
m<5=100O,
m<6=40O

7. Rhombuses
and
Squares

8. Rhombus ≡ ≡ Definition:
A rhombus is a quadrilateral with four congruent sides.
Is a rhombus a parallelogram? ≡
Yes, since opposite sides are congruent. ≡
Therefore…
Opposite sides are parallel and congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.

9. Properties of a Rhombus
Theorem:
The diagonals of a rhombus are perpendicular.
Theorem:
Each diagonal of a rhombus bisects opposite angles.
Note:
The small triangles are RIGHT and CONGRUENT!

10. 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

11. Square
Definition:
A square is a quadrilateral with four congruent angles and four congruent sides.
Is a square a parallelogram?
Yes, since both opposite sides and opposite angles are congruent.
It is also a rectangle and a rhombus.
Therefore…
Opposite sides are parallel and congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
Plus:
Diagonals are congruent and and perpendicular.
Diagonals bisect opposite angles.

12. 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°

13. Trapezoids
and Kites

14. Trapezoid Definition:
A quadrilateral with exactly one pair of opposite sides parallel.
The parallel sides are called bases and the non-parallel sides are called legs.
Consecutive angles between the bases are supplementary.
Base
Trapezoid
Leg
Leg
Base

15. Midsegment of a Trapezoid
The midsegment of a trapezoid is the segment that joins the midpoints of the legs.
Theorem - The midsegment of a trapezoid is parallel to the bases.
Theorem - The length of the midsegment is one-half the sum of the lengths of the bases.
Midsegment

16. Trapezoids – Examples…
Label trapezoid RSTV with bases RS and TV.
RS_____ TV
If mTSR = 75o, then mSTV = ________
Find the midpoint of ST and label it B.
Find the midpoint of RV and label it C. Connect B and C
to create the ____________.
BC is parallel to ____________.
If RS = 8 and TV = 12, then BC = ___. R S C B
105o V T
midsegment RS
and TV 10

17. 6. Find the value of x and m 7. Solve for y 8 (15y – 9)° m 63° x°14 x°
63° m
(90 – 4y)°
(15y – 9)°

18. Isosceles Trapezoid Definition: A trapezoid with congruent legs.

19. 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

20. IscocelesTrapezoids – Examples…
Solve for the missing values.
4x -5
6x - 9 J M K L
3y - 5
7 + y
JL = 4x - 2
MK = 3x + 8
57 y x
x = 57o
4x – 5 = 6x - 9
– 5 = 2x - 9
57 + y = 180
4 = 2x
y = 123o
2 = x

21. Kite
Definition:
A quadrilateral with two distinct pairs of consecutive congruent sides.
Note: opposite sides are NOT congruent!
Theorem:
Diagonals of a kite are
perpendicular.
Theorem:
Exactly one pair of
opposite angles is
congruent.
Note: the congruent angles are created by the
noncongruent adjacent sides.
The pair of opposite angles not congruent is bisected by the diagonal.

22. Kites – Examples… Solve for the missing values. x 21 15 y g + 35 = 90
28˚ g˚ h˚
35˚ x 21 15 y
g + 35 = 90
x = 21
h + 28 = 90
g = 65
h = 62
y = 15

23. Flow Chart Kite Quadrilaterals Parallelogram Rectangle Rhombus Square
Trapezoid
Rhombus
Rectangle
Isosceles
Trapezoid
Square