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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).
- Each side intersects exactly two other sides, one at each endpoint.
- Has vertex/vertices.

Polygons are named by the number of sides they have. Fill in the blank.

Quadrilateral

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

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

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

- 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

- 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

- 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

- 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

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

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

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 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.

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..

- 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.

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

- In class: pp. 342-343 # 1-8 all
- Homework: pp.342-344 #10-18 even, 26, 37

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)

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.

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

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

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 quadrilateralB,D

A,B,C,D

A,B,C,D

B,D

C,D

C

- 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

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.

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

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 not congruent.

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

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

Summarizing Properties of Quadrilaterals

Quadrilateral

Kite

Parallelogram

Trapezoid

Isosceles Trapezoid

Rhombus

Rectangle

Square

Identifying Quadrilaterals

- Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition?

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.

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 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

- 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 a trapezoid with vertices at A(0,0), B(2,4), C(6,4), and D(9,0).

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

Review 1

- A polygon with 7 sides is called a ____.

A) nonagon

B) dodecagon

C) heptagon

D) hexagon

E) decagon

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

- 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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