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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.


Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon


Concave vs convex
Concave vs. Convex in the blank.

  • 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 in the blank.

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

Convex polygon

Concave polygon


  • A polygon is in the blank.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.


Decide whether the polygon is regular
Decide whether the polygon is regular. in the blank.

)

))

)

)

)

)

))

)

)


diagonals


Interior angles of a quadrilateral theorem
Interior Angles of a Quadrilateral Theorem in the blank.

  • 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 in the blank.

  • 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 in the blank.

C

65°

D

55°

123°

B

A


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 in the blank.

  • 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 in the blank.

  • 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.


6 2 properties of parallelograms

6.2 Properties of Parallelograms in the blank.

Geometry

Spring 2014


Objective dfa hw
Objective/DFA/HW in the blank.

  • 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 in the blank.

  • 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 in the blank.

  • 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 in the blank.

  • 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 in the blank.

  • PQRS is a parallelogram. Find the value of h.

P

Q

3h

120°

S

R


Theorems2
Theorems in the blank.

  • 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 in the blank.

  • FGHJ is a parallelogram. Find the unknown length.

    • JH

    • JK

5

5

F

G

3

3

K

J

H


Examples
Examples in the blank.

  • 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. in the blank.

  • 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 °


X

2x-8

Y+10

Y+2


Theorem 6 7
Theorem 6.7 in the blank.

  • If 3 (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.


6 3 proving quadrilaterals are parallelograms

6.3 Proving Quadrilaterals are Parallelograms in the blank.

Geometry

Spring 2014


Objective dfa hw1
Objective/DFA/HW in the blank.

  • Objectives:

    • You will determine whether a quadrialteral is a parallelogram.

  • DFA:

    • pp.372 #12

  • HW:

    • pp.372-374 (2-28 even, 36-44 all)


Review
Review in the blank.


Using properties of parallelograms2
Using properties of parallelograms. in the blank.

  • 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~ in the blank.

  • Show that A(2,0), B(3,4), C(-2,6), and D(-3,2) are the vertices of parallelogram by using method 1.




Proving quadrilaterals are parallelograms
Proving quadrilaterals are parallelograms 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
.. continued.. C(6,6), and D(0,7) is a parallelogram using method 3.

  • Show that the diagonals bisect each other

  • Show that one pair of opposite sides are congruent and parallel.



Example 4 p 341
Example 4 – p.341 C(6,6), and D(0,7) is a parallelogram using method 3.

  • Show that A(2,-1), B(1,3), C(6,5), and D(7,1) are the vertices of a parallelogram.


Assignments
Assignments C(6,6), and D(0,7) is a parallelogram using method 3.

  • In class: pp. 342-343 # 1-8 all

  • Homework: pp.342-344 #10-18 even, 26, 37


6 4 rhombuses rectangles and squares

6.4 Rhombuses, Rectangles, and Squares C(6,6), and D(0,7) is a parallelogram using method 3.

Geometry

Spring 2014


Objective dfa hw2
Objective/DFA/HW C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • Rhombus

    • A rhombus is a parallelogram with four congruent sides.


Special parallelograms1
Special Parallelograms C(6,6), and D(0,7) is a parallelogram using method 3.

  • Rectangle

    • A rectangle is a parallelogram with four right angles.


Special parallelogram
Special Parallelogram C(6,6), and D(0,7) is a parallelogram using method 3.

  • Square

    • A square is a parallelogram with four congruent sides and four right angles.


Corollaries
Corollaries C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

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 C(6,6), and D(0,7) is a parallelogram using method 3.

Geometry

Spring 2014


  • C(6,6), and D(0,7) is a parallelogram using method 3.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 C(6,6), and D(0,7) is a parallelogram using method 3.

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 C(6,6), and D(0,7) is a parallelogram using method 3.

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

B

A

D

C


Pqrs is an isosceles trapezoid find m p m q and m r
PQRS is an isosceles trapezoid. Find m<P, m<Q, and m<R. C(6,6), and D(0,7) is a parallelogram using method 3.

S

R

>

50°

>

P

Q


Isosceles trapezoid1
Isosceles Trapezoid C(6,6), and D(0,7) is a parallelogram using method 3.

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

B

A

D

C


Midsegment theorem for trapezoid
Midsegment Theorem for Trapezoid C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

  • Find the side length.

J

12

H

K

12

14

12

G


6 6 special quadrilaterals

6.6 Special Quadrilaterals C(6,6), and D(0,7) is a parallelogram using method 3.

Geometry

Spring 2014


Summarizing properties of quadrilaterals
Summarizing Properties of Quadrilaterals C(6,6), and D(0,7) is a parallelogram using method 3.

Quadrilateral

Kite

Parallelogram

Trapezoid

Isosceles Trapezoid

Rhombus

Rectangle

Square


Identifying quadrilaterals
Identifying Quadrilaterals C(6,6), and D(0,7) is a parallelogram using method 3.

  • 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 C(6,6), and D(0,7) is a parallelogram using method 3.

always has the given property.

X

X

X

X

X

X

X

X

X

X

X

X



Assignments1
Assignments true, explain why. If it is false, sketch a counterexample.

  • 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)


6 7 areas of triangles and quadrilaterals

6.7 Areas of Triangles and Quadrilaterals true, explain why. If it is false, sketch a counterexample.

Geometry

Spring 2014


Area postulates
Area Postulates true, explain why. If it is false, sketch a counterexample.

  • 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.


Area true, explain why. If it is false, sketch a counterexample.

  • Rectangle: A = bh

  • Parallelogram: A = bh

  • Triangle: A = ½ bh

  • Trapezoid: A = ½ h(b1+b2)

  • Kite: A = ½ d1d2

  • Rhombus: A = ½ d1d2


C

7

4

6

L

B

A

5



Find the area of the figures
Find the area of the figures. B(2,4), C(6,4), and D(9,0).

4

L

L

L

L

4

4

2

L

L

L

L

4

5

8

12


Find the area of abcd
Find the area of ABCD. B(2,4), C(6,4), and D(9,0).

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. B(2,4), C(6,4), and D(9,0).

  • 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. B(2,4), C(6,4), and D(9,0).

  • 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


A

10

D

B

C


Ch 6 review

Ch 6 Review B(2,4), C(6,4), and D(9,0).

Geometry

Spring 2014


Review 1
Review 1 B(2,4), C(6,4), and D(9,0).

  • A polygon with 7 sides is called a ____.

    A) nonagon

    B) dodecagon

    C) heptagon

    D) hexagon

    E) decagon


Review 2
Review 2 B(2,4), C(6,4), and D(9,0).

  • Find m<A

    A) 65°

    B) 135°

    C) 100°

    D) 90°

    E) 105°

B

A

165°

C

30°

65°

D


Review 3
Review 3 B(2,4), C(6,4), and D(9,0).

  • 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 B(2,4), C(6,4), and D(9,0).

  • 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 B(2,4), C(6,4), and D(9,0).

  • 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 B(2,4), C(6,4), and D(9,0).

  • 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 B(2,4), C(6,4), and D(9,0).

  • The quadrilateral below is most specifically a __________.

    A) rhombus

    B) rectangle

    C) kite

    D) parallelogram

    E) trapezoid


Review 8
Review 8 B(2,4), C(6,4), and D(9,0).

  • 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 B(2,4), C(6,4), and D(9,0).

  • 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 B(2,4), C(6,4), and D(9,0).

  • Sketch a concave pentagon.

  • Sketch a convex pentagon.


Review 11
Review 11 B(2,4), C(6,4), and D(9,0).

  • 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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