UNIT 4: POLYGONS

LESSON 1: ANGLES OF POLYGONS AND

QUADRILATERALS

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

Pattern Recognition:

Polygon Interior Angles Sum: The sum of the interior angle measures of an n-sided convex polygon is

(n – 2 ) *180

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

EXTERIOR ANGLE SUM

360The sum of the exterior angles is

The measure of one exterior angle is

360

n

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.

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.

Find the measure of each interior angle.

Find the measure of each interior angle.

Answer:

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

FIND X

FIND X

A. 10

B. 12

C. 14

D. 15

14

PRACTICE…..

360 36045

8n

1. Sum of the measures of the interior angles of a 11-gon is

(n – 2)180° (11 – 2)180 ° 1620

2. The measure of an exterior angle of a regular octagon is

3. The number of sides of regular polygon with exterior angle 72 ° is

4. The measure of an interior angle of a regular polygon with 30 sides

360 3605

72n n

exterior angle

15

IsoscelesTrapezoid

Quadrilaterals

Rectangle

Parallelogram

Rhombus

Square

Flow Chart

Trapezoid

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, the figure above can be either

ABCD or ADCB.

Lesson 6-1: Parallelogram 16

Definition: A quadrilateral whose opposite sides are parallel.

Symbol: a smaller versionof a parallelogram

Naming:

CB

A D

PROPERTIES OF PARALLELOGRAM

A C and B D

180 180

180 180

m A m B and m A m D

m B m C and m C m D

17

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 .

A B

CDP

EXAMPLES

1. Draw HKLP.

2. 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 =____.

6. Draw the diagonals with their point of intersection labeled M.

7. 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 = _____ .

18

H K

LP

PL KL

P

P or K

115° 65 115°

M

5 units

14 units7.5 units

36; (Alternate interior angles are congruent.)

Proving Quadrilaterals as Parallelograms

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram .

Theorem 1:

H G

E FIf one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .

Theorem 2:

If EF GH; FG EH, then Quad. EFGH is a parallelogram.

If EF GH and EF || HG, then Quad. EFGH is a parallelogram.

Theorem:

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 3:

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram .

Theorem 4:

H G

EF

M

,If H F and E G

then Quad. EFGH is a parallelogram.

intIf M is the midpo of EG and FH

then Quad. EFGH is a parallelogram. EM = GM and HM = FM

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 .

EXAMPLES ……

Find the value of x and y that ensures the quadrilateral is a parallelogram.

Example 1:

6x4x+8

y+2

2y

6x = 4x+8

2x = 8

x = 4 units

2y = y+2

y = 2 unit

Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram.

120° 5y°

(2x + 8)°

2x + 8 = 120

2x = 112

x = 56 units

5y + 120 = 180

5y = 60

y = 12 units

EXAMPLE 3

A. yes

B. no

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.

Lesson 6-3: Rectangles 24

RECTANGLES

• Opposite sides are parallel.• Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• Diagonals bisect each other.

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.

25

PROPERTIES OF RECTANGLES

Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles.

If a parallelogram is a rectangle, then its diagonals are congruent.

E

D C

BA

Theorem:

Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle.

Lesson 6-3: Rectangles 26

EXAMPLES…….

1. If AE = 3x +2 and BE = 29, find the value of x.

2. If AC = 21, then BE = _______.

3. If m<1 = 4x and m<4 = 2x, find the value of x.

4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.

m<1=50, m<3=40, m<4=80, m<5=100, m<6=40

10.5 units

x = 9 units

x = 18 units

6

54

321

E

D C

BA

EXAMPLE 5

A. x = 1

B. x = 3

C. x = 5

D. x = 10

Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x.

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.

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.

1. If AB = 9, then AD = ______.

2. If m<1 = 65, the m<2 = _____.

3. m<3 = ______.

4. If m<ADC = 80, the m<DAB = ______.

5. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.

54

3

21E

D C

BA9 units

65°

90°

100°

10

32

SQUARE

• 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.

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.

33

SQUARES – EXAMPLES…...Given: ABCD is a square. Complete the following.

1. If AB = 10, then AD = _____ and DC = _____.

2. If CE = 5, then DE = _____.

3. m<ABC = _____.

4. m<ACD = _____.

5. m<AED = _____.

8 7 65

4321

E

D C

BA

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 a quadrilateral is both a rectangle and a rhombus, then it is a square.

Theorem: If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Lesson 6-5: Trapezoid & Kites 35

TRAPEZOID

A quadrilateral with exactly one pair of parallel sides.Definition:

BaseLeg

An Isosceles trapezoid is a trapezoid with congruent legs.

Trapezoid

The parallel sides are called bases and the non-parallel sides are called legs.

Isosceles trapezoid

Lesson 6-5: Trapezoid & Kites 36

PROPERTIES OF ISOSCELES TRAPEZOID

A B and D C

2. The diagonals of an isosceles trapezoid are congruent.

1. Both pairs of base angles of an isosceles trapezoid are congruent.

A B

CD

Base Angles

AC DB

Lesson 6-5: Trapezoid & Kites 37

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

1b

2b

1 2

1( )

2median b b

Median of a Trapezoid