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6.1 Polygons. Geometry. Objectives/DFA/HW. Objectives: You will solve problems using the interior & exterior angle-sum theorems. DFA: pp.356-357 #16 & #30 HW: pp.356-358 (2-44 even). What is polygon?. Formed by three or more segments (sides).

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6 1 polygons

6.1 Polygons

Geometry

objectives dfa hw
Objectives/DFA/HW
  • Objectives:
    • You will solve problems using the interior & exterior angle-sum theorems.
  • DFA:
    • pp.356-357 #16 & #30
  • HW:
    • pp.356-358 (2-44 even)
what is polygon
What is polygon?
  • Formed by three or more segments (sides).
  • Each side intersects exactly two other sides, one at each endpoint.
  • Has vertex/vertices.
slide4
Polygons are named by the number of sides they have. Fill in the blank.

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

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

Convex polygon

Concave polygon

slide7
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.
slide9
A Diagonal of a polygon is a segment that joins two nonconsecutive vertices.

diagonals

interior angles of a quadrilateral theorem
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

example1
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
Find m<A

C

65°

D

55°

123°

B

A

slide13
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 °

theorem 6 1 polygon angle sum theorem
Theorem 6-1 – Polygon Angle-Sum Theorem
  • The sum of the measures of the interior angles of an n-gon is (n-2)180.
  • Ex. What is the sum of the interior angle measures of a heptagon?
theorem 6 2 polygon exterior angle sum theorem
Theorem 6-2 Polygon Exterior Angle-Sum Theorem
  • The sum of the measures of the exterior angles of polygon, one at each vertex is 360o.
  • For the petagon
    • m<1+m<2+m<3+m<4+m<5=360
    • Ex. What is the measure of each

angle of an octagon.

objective dfa hw
Objective/DFA/HW
  • Objectives:
    • You will use properties (angles & sides) of parallelograms & relationships among diagonals to solve problems relating to parallelograms.
  • DFA:
    • pp.364 #16 & #22
  • HW:
    • pp.363-366 (2-40 even)
theorems
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

theorems1
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
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
Using Algebra with Parallelograms
  • PQRS is a parallelogram. Find the value of h.

P

Q

3h

120°

S

R

theorems2
Theorems
  • If a quadrilateral is a parallelogram, then its diagonals bisect each other.

R

Q

M

P

S

using properties of parallelograms1
Using properties of parallelograms
  • FGHJ is a parallelogram. Find the unknown length.
    • JH
    • JK

5

5

F

G

3

3

K

J

H

examples
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
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 °

theorem 6 7
Theorem 6.7
  • If 3 (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
objective dfa hw1
Objective/DFA/HW
  • Objectives:
    • You will determine whether a quadrialteral is a parallelogram.
  • DFA:
    • pp.372 #12
  • HW:
    • pp.372-374 (2-28 even, 36-44 all)
using properties of parallelograms2
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
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.
slide33
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.
slide34
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.
proving quadrilaterals are parallelograms
Proving quadrilaterals are parallelograms
  • 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
.. continued..
  • Show that the diagonals bisect each other
  • Show that one pair of opposite sides are congruent and parallel.
slide37
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.
example 4 p 341
Example 4 – p.341
  • Show that A(2,-1), B(1,3), C(6,5), and D(7,1) are the vertices of a parallelogram.
assignments
Assignments
  • In class: pp. 342-343 # 1-8 all
  • Homework: pp.342-344 #10-18 even, 26, 37
objective dfa hw2
Objective/DFA/HW
  • Objectives:
    • You will determine whether a parallelogram is a rhombus, rectangle, or a square & you will solve problems using properties of special parallelograms.
  • DFA:
    • pp.379 #12
  • HW:
    • pp.379-382 (1-27all)
review1
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
Special Parallelograms
  • Rhombus
    • A rhombus is a parallelogram with four congruent sides.
special parallelograms1
Special Parallelograms
  • Rectangle
    • A rectangle is a parallelogram with four right angles.
special parallelogram
Special Parallelogram
  • Square
    • A square is a parallelogram with four congruent sides and four right angles.
corollaries
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.
example2
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

review2
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

example3
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
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
Theorem of rectangle
  • A parallelogram is a rectangle if and only if its diagonals are congruent.

A

B

D

C

match the properties of a quadrilateral
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

6.5 Trapezoid and Kites

Geometry

Spring 2014

slide54

• Objective

    • You will determine whether a parallelogram is a kite or a trapezoid.
  • • DFA –
    • P.387 # 18
  • o HW –
    • p.386-388 (2-34 even)
let s define trapezoid
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
Isosceles Trapezoid
  • If a trapezoid is isosceles, then each pair of base angles is congruent.

B

A

D

C

isosceles trapezoid1
Isosceles Trapezoid
  • A trapezoid is isosceles if and only if its diagonals are congruent.

B

A

D

C

midsegment theorem for trapezoid
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 bases. It’s the average of the lengths of the bases.

C

B

N

M

A

D

examples1
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

examples2
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

review3
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

slide63
Kite
  • A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
theorems about kites
Theorems about Kites
  • 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

example4
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

example5
Example
  • Find the side length.

J

12

H

K

12

14

12

G

6 6 special quadrilaterals

6.6 Special Quadrilaterals

Geometry

Spring 2014

summarizing properties of quadrilaterals
Summarizing Properties of Quadrilaterals

Quadrilateral

Kite

Parallelogram

Trapezoid

Isosceles Trapezoid

Rhombus

Rectangle

Square

identifying quadrilaterals
Identifying Quadrilaterals
  • Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition?
slide70
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

slide71
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.
assignments1
Assignments
  • pp. 359-361 # 3-24, 28-34, 37-39 (odd in class; even for homework)
  • pp. 367-368 # 16-41 (odd in class; even for homework)
area postulates
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.
  • Area Addition Postulate
    • The area of a region is the sum of the areas of its non-overlapping parts.
slide75
Area
  • Rectangle: A = bh
  • Parallelogram: A = bh
  • Triangle: A = ½ bh
  • Trapezoid: A = ½ h(b1+b2)
  • Kite: A = ½ d1d2
  • Rhombus: A = ½ d1d2
find the area of abcd
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.
  • 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.
  • 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

slide82
The area of the kite is160.
  • Find the length of BD.

A

10

D

B

C

ch 6 review

Ch 6 Review

Geometry

Spring 2014

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

A) nonagon

B) dodecagon

C) heptagon

D) hexagon

E) decagon

review 2
Review 2
  • Find m<A

A) 65°

B) 135°

C) 100°

D) 90°

E) 105°

B

A

165°

C

30°

65°

D

review 3
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
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
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
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
Review 7
  • The quadrilateral below is most specifically a __________.

A) rhombus

B) rectangle

C) kite

D) parallelogram

E) trapezoid

review 8
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
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
Review 10
  • Sketch a concave pentagon.
  • Sketch a convex pentagon.
review 11
Review 11
  • What type of quadrilateral is ABCD? Explain your reasoning.

D

120°

A

60°

C

120°

Isosceles Trapezoid

Isosceles : AD = BC

Trapezoid : AB ll CD

60°

B

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