6.1 Polygons

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# 6.1 Polygons - PowerPoint PPT Presentation

6.1 Polygons. Day 1 Part 1 CA Standards 7.0, 12.0, 13.0. Warmup. Solve for the variables. 1. 10 + 8 + 16 + A = 36 2. 6 + 15 + 9 + 3B = 36 3. 10 + 8 + 2X + 2X = 36 4. 4R + 10 + 108 + 67 + 3R = 360. What is polygon?. Formed by three or more segments (sides).

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### 6.1 Polygons

Day 1 Part 1

CA Standards 7.0, 12.0, 13.0

Warmup
• Solve for the variables.
• 1. 10 + 8 + 16 + A = 36
• 2. 6 + 15 + 9 + 3B = 36
• 3. 10 + 8 + 2X + 2X = 36
• 4. 4R + 10 + 108 + 67 + 3R = 360
What is polygon?
• Formed by three or more segments (sides).
• Each side intersects exactly two other sides, one at each endpoint.
• Has vertex/vertices.

Pentagon

Hexagon

Heptagon

Octagon

Concave vs. Convex
• Convex: if no line that contains a side of the polygon contains a point in the interior of the polygon.
• Concave: if a polygon is not convex.

interior

Example
• Identify the polygon and state whether it is convex or concave.

Convex polygon

Concave polygon

A polygon is equilateral if all of its sides are congruent.
• A polygon is equiangular if all of its interior angles are congruent.
• A polygon is regular if it is equilateral and equiangular.
A Diagonal of a polygon is a segment that joins two nonconsecutive vertices.

diagonals

Interior Angles of a Quadrilateral Theorem
• The sum of the measures of the interior angles of a quadrilateral is 360°.

B

m<A + m<B + m<C + m<D = 360°

C

A

D

Example
• Find m<Q and m<R.

x + 2x + 70° + 80° = 360°

3x + 150 ° = 360 °

3x = 210 °

x = 70 °

Q

x

2x°

R

80°

P

70°

m< Q = x

m< Q = 70 °

m<R = 2x

m<R = 2(70°)

m<R = 140 °

S

Find m<A

C

65°

D

55°

123°

B

A

Use the information in the diagram to solve for j.

60° + 150° + 3j ° + 90° = 360°

210° + 3j ° + 90° = 360°

300° + 3j ° = 360 °

3j ° = 60 °

j = 20

60°

150°

3j °

### 6.2 Properties of Parallelograms

Day 1 Part 2

CA Standards 4.0, 7.0, 12.0, 13.0, 16.0, 17.0

Theorems
• If a quadrilateral is a parallelogram, then its opposite sides are congruent.
• If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Q

R

S

P

Theorems
• If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

m<P + m<Q = 180°

m<Q + m<R = 180°

m<R + m<S = 180°

m<S + m<P = 180°

Q

R

S

P

Using Properties of Parallelograms
• PQRS is a parallelogram. Find the angle measure.
• m< R
• m< Q

Q

70 °

R

70 ° + m < Q = 180 °

m< Q = 110 °

70°

P

S

Using Algebra with Parallelograms
• PQRS is a parallelogram. Find the value of h.

P

Q

3h

120°

S

R

Theorems
• If a quadrilateral is a parallelogram, then its diagonals bisect each other.

R

Q

M

P

S

Using properties of parallelograms
• FGHJ is a parallelogram. Find the unknown length.
• JH
• JK

5

5

F

G

3

3

K

J

H

Examples
• Use the diagram of parallelogram JKLM. Complete the statement.

LM

K

L

NK

<KJM

N

<LMJ

NL

MJ

J

M

Find the measure in parallelogram LMNQ.
• LM
• LP
• LQ
• QP
• m<LMN
• m<NQL
• m<MNQ
• m<LMQ

18

8

L

M

9

110°

10

10

9

P

70°

8

32°

70 °

Q

N

18

110 °

32 °

### 6.3 Proving Quadrilaterals are Parallelograms

Day 2 Part 1

CA Standards 4.0, 7.0, 12.0, 17.0

Warmup
• Find the slope of AB.
• A(2,1), B(6,9)

m=2

• A(-4,2), B(2, -1)

m= - ½

• A(-8, -4), B(-1, -3)

m= 1/7

Using properties of parallelograms.
• Method 1

Use the slope formula to show that opposite sides have the same slope, so they are parallel.

• Method 2

Use the distance formula to show that the opposite sides have the same length.

• Method 3

Use both slope and distance formula to show one pair of opposite side is congruent and parallel.

Let’s apply~
• Show that A(2,0), B(3,4), C(-2,6), and D(-3,2) are the vertices of parallelogram by using method 1.
Show that the quadrilateral with vertices A(-3,0), B(-2,-4), C(-7, -6) and D(-8, -2) is a parallelogram using method 2.
Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(6,6), and D(0,7) is a parallelogram using method 3.
• Show that both pairs of opposite sides are parallel.
• Show that both pairs of opposite sides are congruent.
• Show that both pairs of opposite angles are congruent.
• Show that one angle is supplementary to both consecutive angles.
.. continued..
• Show that the diagonals bisect each other
• Show that one pair of opposite sides are congruent and parallel.
Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(6,6), and D(0,7) is a parallelogram using method 3.

### 6.4 Rhombuses, Rectangles, and Squares

Day 2 Part 2

CA Standards 4.0, 7.0, 12.0, 17.0

Review
• Find the value of the variables.

p

h

52°

(2p-14)°

50°

68°

p + 50° + (2p – 14)° = 180°

p + 2p + 50° - 14° = 180°

3p + 36° = 180°

3p = 144 °

p = 48 °

52° + 68° + h = 180°

120° + h = 180 °

h = 60°

Special Parallelograms
• Rhombus
• A rhombus is a parallelogram with four congruent sides.
Special Parallelograms
• Rectangle
• A rectangle is a parallelogram with four right angles.
Special Parallelogram
• Square
• A square is a parallelogram with four congruent sides and four right angles.
Corollaries
• Rhombus corollary
• A quadrilateral is a rhombus if and only if it has four congruent sides.
• Rectangle corollary
• A quadrilateral is a rectangle if and only if it has four right angles.
• Square corollary
• A quadrilateral is a square if and only if it is a rhombus and a rectangle.
Example
• PQRS is a rhombus. What is the value of b?

Q

2b + 3 = 5b – 6

9 = 3b

3 = b

P

2b + 3

R

S

5b – 6

Review
• In rectangle ABCD, if AB = 7f – 3 and CD = 4f + 9, then f = ___
• 1
• 2
• 3
• 4
• 5

7f – 3 = 4f + 9

3f – 3 = 9

3f = 12

f = 4

Example
• PQRS is a rhombus. What is the value of b?

Q

3b + 12 = 5b – 6

18 = 2b

9 = b

P

3b + 12

R

S

5b – 6

Theorems for rhombus
• A parallelogram is a rhombus if and only if its diagonals are perpendicular.
• A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

L

Theorem of rectangle
• A parallelogram is a rectangle if and only if its diagonals are congruent.

A

B

D

C

The diagonals are congruent

Both pairs of opposite sides are congruent

Both pairs of opposite sides are parallel

All angles are congruent

All sides are congruent

Diagonals bisect the angles

Parallelogram

Rectangle

Rhombus

Square

Match the properties of a quadrilateral

B,D

A,B,C,D

A,B,C,D

B,D

C,D

C

### 6.5 Trapezoid and Kites

Day 3 Part 1

CA Standards 4.0, 7.0, 12.0

Which of these sums is equal to a negative number?

(4) + (-7) + (6)

(-7) + (-4)

(-4) + (7)

(4) + (7)

In the first seven games of the basketball season, Cindy scored 8, 2, 12, 6, 8, 4 and 9 points. What was her mean number of points scored per game?

6

7

8

9

Warmup
Let’s define Trapezoid

base

A

B

>

leg

leg

>

C

D

base

<D AND <C ARE ONE PAIR OF BASE ANGLES.

When the legs of a trapezoid are congruent,

then the trapezoid is an isosceles trapezoid.

Isosceles Trapezoid
• If a trapezoid is isosceles, then each pair of base angles is congruent.

B

A

D

C

Isosceles Trapezoid
• A trapezoid is isosceles if and only if its diagonals are congruent.

B

A

D

C

Midsegment Theorem for Trapezoid
• The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the base.

C

B

N

M

A

D

Examples
• The midsegment of the trapezoid is RT. Find the value of x.

7

R

x

T

x = ½ (7 + 14)

x = ½ (21)

x = 21/2

14

Examples
• The midsegment of the trapezoid is ST. Find the value of x.

8

S

11

T

11 = ½ (8 + x)

22 = 8 + x

14 = x

x

Review

In a rectangle ABCD, if AB = 7x – 3, and CD = 4x + 9, then x = ___

A) 1

B) 2

C) 3

D) 4

E) 5

7x – 3 = 4x + 9

-4x -4x

3x – 3 = 9

+ 3 +3

3x = 12

x = 4

Kite
• A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are congruent.
• If a quadrilateral is a kite, then its diagonals are perpendicular
• If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

B

A

C

L

D

Example
• Find m<G and m<J.

J

Since m<G = m<J,

2(m<G) + 132° + 60° = 360°

2(m<G) + 192° = 360°

2(m<G) = 168°

m<G = 84°

H

132°

60°

K

G

Example
• Find the side length.

J

12

H

K

12

14

12

G

Day 3 Part 2

CA Standards 7.0, 12.0

Kite

Parallelogram

Trapezoid

Isosceles Trapezoid

Rhombus

Rectangle

Square

• Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition?
Copy the chart. Put an X in the box if the shape

always has the given property.

X

X

X

X

X

X

X

X

X

X

X

X

Determine whether the statement is true or false. If it is true, explain why. If it is false, sketch a counterexample.
• If CDEF is a kite, then CDEF is a convex polygon.
• If GHIJ is a kite, then GHIJ is not a trapezoid.
• The number of acute angles in a trapezoid is always either 1 or 2.
Pg. 359 # 3 – 33, 40
• Pg. 368 # 16 – 41

### 6.7 Areas of Triangles and Quadrilaterals

Day 4 Part 1

CA Standard 7.0, 8.0, 10.0

Warmup

1.

2.

3.

Area Postulates
• Area of a Square Postulate
• The area of a square is the square of the length of its sides, or A = s2.
• Area Congruence Postulate
• If two polygons are congruent, then they have the same area.
• The area of a region is the sum of the areas of its non-overlapping parts.
Area
• Rectangle: A = bh
• Parallelogram: A = bh
• Triangle: A = ½ bh
• Trapezoid: A = ½ h(b1+b2)
• Kite: A = ½ d1 d2
• Rhombus: A = ½ d1 d2
Find the area of ABCD.

B

C

ABCD is a parallelogram

Area = bh

= (16)(9)

= 144

9

E

16

A

D

12

Find the area of a trapezoid.
• Find the area of a trapezoid WXYZ with W(8,1), X(1,1), Y(2,5), and Z(5,5).
Find the area of rhombus.
• Find the area of rhombus ABCD.

B

Area of Rhombus

A = ½ d1 d2

= ½ (40)(30)

= ½ (1200)

= 600

15

20

20

A

C

15

25

D

The area of the kite is160.
• Find the length of BD.

A

10

D

B

C

### Ch 6 Review

Day 4 Part 2

Review 1
• A polygon with 7 sides is called a ____.

A) nonagon

B) dodecagon

C) heptagon

D) hexagon

E) decagon

Review 2
• Find m<A

A) 65°

B) 135°

C) 100°

D) 90°

E) 105°

B

A

165°

C

30°

65°

D

Review 3
• Opposite angles of a parallelogram must be _______.

A) complementary

B) supplementary

C) congruent

D) A and C

E) B and C

Review 4
• If a quadrilateral has four equal sides, then it must be a _______.

A) rectangle

B) square

C) rhombus

D) A and B

E) B and C

Review 5
• The perimeter of a square MNOP is 72 inches, and NO = 2x + 6. What is the value of x?

A) 15

B) 12

C) 6

D) 9

E) 18

Review 6
• ABCD is a trapezoid. Find the length of midsegment EF.

A) 5

B) 11

C) 16

D) 8

E) 22

13

A

E

11

B

5

D

F

C

9

Review 7
• The quadrilateral below is most specifically a __________.

A) rhombus

B) rectangle

C) kite

D) parallelogram

E) trapezoid

Review 8
• Find the base length of a triangle with an area of 52 cm2 and a height of 13cm.

A) 8 cm

B) 16 cm

C) 4 cm

D) 2 cm

E) 26 cm

Review 9
• A right triangle has legs of 24 units and 18 units. The length of the hypotenuse is ____.

A) 15 units

B) 30 units

C) 45 units

D) 15.9 units

E) 32 units

Review 10
• Sketch a concave pentagon.
• Sketch a convex pentagon.
Review 11

D

120°

A

60°

C

120°

Isosceles Trapezoid