warm-up 1/31/12 given the number of sides of a regular polygon, find: (a) the sum of the interior...

Warm-up 1/31/12

Given the number of sides of a regular polygon, Find: (a) the sum of the interior angles (b) one interior angle1.5 sides 2. 10 sides

3.What is the sum of the exterior angles of a 13-gon?

4.How many sides does the regular polygon have if the measure of its exterior angle is 18 degrees?

(a) S= (5-2)180 = (3)(180)=540

(b) One interior angle = 540/5 = 108

(a) S= (10-2)180 = (8)(180)=1440

(b) One interior angle =1440/10 = 144

360

360/1820 sides

Homework Answers:

(1). No (2). Yes (3). No (4). Yes (5). No (6). Pentagon (7). Quadrilateral (8). Hexagon Convex Convex Concave

(9). Hexagon (10). Dodecagon Convex Concave

(11). Equilateral (12). Regular (13). Equiangular(14). Regular (15). Neither

Homework Answers:

(16). a. 2340b. 360

(17). 15624

(18). a. 3240b. 360

(19). 162 18

(20). LMNOPpentagon

(21). ABCDEFhexagon

(22). 540

(23). 150

(24). x = 50

4.2 Parallelograms

Essential Questions:

(1). How are quadrilaterals classified by their angles or sides?

(2). How are parallelograms classified? (3). How can we prove a quadrilateral is a parallelogram?

Essential Question #1&2

How are quadrilaterals / parallelograms classified by their angles or sides?

Quadrilaterals are polygons with ______ edges.

The prefix " quad" means _________ and the word "lateral" means ________.

four

fouredges

Below are some examples and non-examples of

quadrilaterals. Circle the figures you believe are

quadrilaterals.

A BE

F

GCH

D

Circle the quadrilaterals that have one pair of parallel edges.

A B CD E

F

G

H

Some quadrilaterals are special because they have parallel edges. If a quadrilateral has one pair of parallel edges it is called a _________.trapezoid

When a figure has two pairs of parallel edges, we say that both pairs of opposite edges are parallel.

A quadrilateral with both pairs of opposite edges parallel is called a ________________.

Circle the quadrilaterals below that you believe are parallelograms.

L

parallelogram

AB C D

EF G H I

J L

• A and H are parallelograms that have four

congruent edges. These parallelograms are called

__________ (plural of __________).

• E, F, and H are parallelograms that have four

right angles. These parallelograms are called

_________.

• Notice that H has both four equal sides and four

right angles. This parallelogram is called a

_______.

rhombi rhombus

rectangles

square

Definitions:

(1). Quadrilateral:

(2). Parallelogram:

(3). Rectangle

(4). Rhombus:

A polygon with exactly 4 edges.

A quadrilateral with 2 pairs ofopposite edges that are parallel.

A quadrilateral with 4 right angles.

A quadrilateral with 4 congruentedges.

Definitions:

(5). Square:

(6). Trapezoid:

A quadrilateral with 4 right anglesand 4 congruent edges.

A quadrilateral with exactly onepair of opposite edges that areparallel.

Quadrilaterals

• In the figures below, place an A in each

quadrilateral, a B in each parallelogram, a C in

each rectangle, a D in each rhombus, an E in

each square and an F in each trapezoid.

Quadrilaterals

• The last letter you wrote in each figure is the

most specific name for that figure. Color the

figures according to their most specific name.

• Quadrilaterals - white, Parallelograms - green, Rectangles - blue, Rhombi - orange, Squares - Yellow, Trapezoids - purple.

QuadrilateralsA

A A A

A

AAA

AA

A

A

BBB

B

BB B

B

C

C

D

D

D

D

E

F

F

C

QuadrilateralsA

A AA

A

AAA

AA

A

A

B

BB

B

BB B

B

C

C

D

D

D

D

E

F

F

. . put our heads together to….

http://wrir4.ucdavis.edu/PHOTOS/CROPS/images/Lettuce%20WideBeds.jpg

Answer Essential Question #1 & 2

How are quadrilaterals / parallelograms classified by their

angles or sides?

Essential Question #3

How can we prove a quadrilateral is a parallelogram?

Now, we are going to find the five special characteristics of all parallelograms:

(1) The first characteristic is based on the definition of a parallelogram:

Both pairs of opposite sides of a parallelogram are ___________.parallel

(2) The second characteristic is based on edge lengths. Record the measure of all the angles in the parallelograms pictured below.

If a figure is a parallelogram then both pairs of opposite edges are _______________.

2.3 cm

2.3 cm

2 cm

2 cm

3.6 cm

3.6 cm

1.9

cm

1.9

cm 1.8

cm

1.8

cm

3.4 cm

3.4 cm

congruent

(3) The third characteristic is based on angle measures. Record the measure of all the angles in the parallelograms below.

69o

69o111o

111o 100o

100o

80o

80o

congruentIf a figure is a parallelogram then both pairs of opposite angles are ____________.

A

D C

B

(4) You will discover the fourth characteristic. Using the parallelograms above, what is special about m A and m B? ___________________ A and B are called consecutive angles.

Here are some other pairs of consecutive angles:B and C C and D D and A

They are supplementary

If a figure is a parallelogram then consecutiveangles are ______________.

(They add up to _____o).

supplementary

180

(5) Now we discover the fifth and final characteristic. about a parallelogram. This characteristic pertains to the diagonals of the parallelogram. The diagonal of a polygon is a segment whose endpoints are nonconsecutive vertices of the polygon.

Draw the diagonals of each parallelogram below. Todo this you simply connect the points A and C with aline segment. To draw the second diagonal youconnect the points B and D with a line segment.

A B

D C

AB

D

C

A B

D C

Label the point where two diagonals intersect as point X. Now, measure the lengths of the followingsegments: AX, XC, BX, XD. Write the lengthnext to each segment.

A B

D C

AB

D

C

A B

D C

XX X

1.21.9

1.92.1 1.9 1.9

1.9 1.92.11.9

1.91.2

The segment AX is congruent to segment _____.That means that X is the _____________ ofsegment AC.

XCmidpoint

The segment BX is congruent to segment _____.That means that X is the _____________ ofsegment BD.

This conclusion leads us to the fifth characteristicof a parallelogram:

DXmidpoint

The diagonals of a parallelogram _________each other.

bisect

Conclusion, list the five characteristics ofall parallelograms:

1. ___________________________________2. ___________________________________3. ___________________________________4. ___________________________________

5. ___________________________________

Opposite edges are parallel.

Opposite edges are congruent.Opposite angles are congruent.Consecutive Interior Angles are SupplementaryDiagonals bisect each other

Remember these characteristics apply to all parallelograms, rectangles, rhombi and squares.

. . put our heads together to….

http://wrir4.ucdavis.edu/PHOTOS/CROPS/images/Lettuce%20WideBeds.jpg

Answer Essential Question #3

How can we prove a quadrilateral is a parallelogram?