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 angle 1.5 sides2. 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 = /18 20 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 b. 360 (17) (18). a b. 360 (19) (20). LMNOP pentagon (21). ABCDEF hexagon (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 edges

Below are some examples and non-examples of quadrilaterals. Circle the figures you believe are quadrilaterals. AB E F G C H D

Circle the quadrilaterals that have one pair of parallel edges. A B C D 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 A B C D E F G HI 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 _______. rhombirhombus rectangles square

Definitions: (1). Quadrilateral: (2). Parallelogram: (3). Rectangle (4). Rhombus: A polygon with exactly 4 edges. A quadrilateral with 2 pairs of opposite edges that are parallel. A quadrilateral with 4 right angles. A quadrilateral with 4 congruent edges.

Definitions: (5). Square: (6). Trapezoid: A quadrilateral with 4 right angles and 4 congruent edges. A quadrilateral with exactly one pair of opposite edges that are parallel.

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.

Quadrilaterals A A A A A A A A A A A A B B B B B B B B C C D D D D E F F C

A A A A A A A A A A A A B B B B B B B B C C D D D D E F F

.. put our heads together to…. 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 cm 3.6 cm 1.9 cm 1.8 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. 69 o 111 o 100 o 80 o congruent If 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 consecutive angles 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. To do this you simply connect the points A and C with a line segment. To draw the second diagonal you connect the points B and D with a line segment. A B D C A B D C A B D C

Label the point where two diagonals intersect as point X. Now, measure the lengths of the following segments: AX, XC, BX, XD. Write the length next to each segment. A B D C A B D C A B D C X X X

The segment AX is congruent to segment _____. That means that X is the _____________ of segment AC. XC midpoint The segment BX is congruent to segment _____. That means that X is the _____________ of segment BD. This conclusion leads us to the fifth characteristic of a parallelogram: DX midpoint The diagonals of a parallelogram _________ each other. bisect

Conclusion, list the five characteristics of all parallelograms: 1. ___________________________________ 2. ___________________________________ 3. ___________________________________ 4. ___________________________________ 5. ___________________________________ Opposite edges are parallel. Opposite edges are congruent. Opposite angles are congruent. Consecutive Interior Angles are Supplementary Diagonals bisect each other Remember these characteristics apply to all parallelograms, rectangles, rhombi and squares.

.. put our heads together to…. Answer Essential Question #3 How can we prove a quadrilateral is a parallelogram?