1. Unit 4: Polygons

2. Lesson 1: Angles of Polygons and Quadrilaterals

3. Diagonal: a diagonal of a polygon is a segment that connects any two nonconsecutive vertices.

4. Pattern Recognition:
Polygon Interior Angles Sum: The sum of the interior angle measures of an n-sided convex polygon is
(n – 2 ) *180

5. A decorative window is designed to have the shape of a regular octagon
A decorative window is designed to have the shape of a regular octagon. Find the sum of the measures of the interior angles of the octagon.
Answer: 1080

6. EXTERIOR ANGLE SUM The sum of the exterior angles is
The measure of one exterior angle is

7. The measure of an interior angle of a regular polygon is 135
The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.
Answer: The polygon has 8 sides.

8. The measure of an interior angle of a regular polygon is 144
The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.
Answer: The polygon has 10 sides.

9. Find the measure of each interior angle.

10. Find the measure of each interior angle.
Answer:

11. Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ.

12. Find x
----- Meeting Notes (10/3/12 15:23) -----
31x - 12 = 360

13. Find x
A. 10
B. 12
C. 14
D. 15

14. pRACTICE….. Sum of the measures of the interior angles of a 11-gon is
The measure of an exterior angle of a regular octagon is
The number of sides of regular polygon with exterior angle 72 ° is
The measure of an interior angle of a regular polygon with 30 sides

15. Flow Chart Quadrilaterals Parallelogram Rhombus Rectangle Square
Trapezoid
Parallelogram
Rhombus
Isosceles
Trapezoid
Rectangle
Square

16. 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

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

18. Examples H K 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.)

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 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
120°
5y°
(2x + 8)°

23. Example 3
Use the Distance Formula to determine if A(3, 7), B(9, 10), C(10, 6), D(4, 3) are the vertices of a parallelogram.
A. yes
B. no

24. 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

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.

26. 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

27. Example 5
Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x.
A. x = 1
B. x = 3
C. x = 5
D. x = 10

28. ART Some artists stretch their own canvas over wooden frames
ART Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches, explain how an artist can be sure that the frame is rectangular.
Since AB = CD, DA = BC, and AC = BD, AB  CD, DA  BC, and AC  BD.
Answer: Because AB  CD and DA  BC, ABCD is a parallelogram. Since AC and BD are congruent diagonals in parallelogram ABCD, it is a rectangle.

29. Rhombus ≡ ≡ 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

30. Properties of a Rhombus
Theorem:
The diagonals of a rhombus are perpendicular.
Theorem:
Each diagonal of a rhombus bisects a pair of opposite angles.

31. 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

32. 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.

33. Squares – Examples…... 10 units 10 units 5 units 90° 45° 90°
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°

34. Conditions of Rhombi and Squares
Theorem:
If the diagonals of a parallelograms are perpendicular , then the parallelogram is a rhombus.
Theorem:
If one diagonal bisects a pair of opposite angles then the parallelogram is a rhombus.
Theorem:
If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.
Theorem:
If a quadrilateral is both a rectangle and a rhombus, then it is a square.

35. 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

36. 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

37. 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