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