Any questions about the material so far? About the exercises?
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2 Any questions about the material so far? About the exercises? Here is a question for you. In the diagram on the board, DE is parallel to AC, DB = 4, AB = 9 and BE = 8. What is the length EC?
3 Polygons Definitions: A polygon is a closed plane figure whose sides are line segments. Points where two segments meet are called vertices. The angles formed inside the polygon are called interior angles. A polygon is regular if all its sides and interior angles are congruent. Note: A polygon with three sides is a triangle, with four sides it is called a quadrilateral; with fives sides, it is a pentagon.
4 NUMBER OF SIDES POLYGON 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon
5 Theorem: The sum of the interior angles of a n-gon is Proof: (n - 2)180 Pick a vertex and then draw a line segment to each of the other vertices as shown above forming some triangles. Note that that the first and last triangles formed contain two sides of the polygon but each of the others contain only one side. Thus there are two fewer triangles than there are sides; i.e., if there are n sides, then there are n - 2 triangles formed. Since each triangle has 180 degrees and there are n - 2 of them, the total for the polygon is (n - 2)180.
6 The sum of the interior angles of a quadrilateral is (4-2)180 = 360 degrees. The sum of the interior angles of a pentagon is (5-2)180 = 540 degrees. Etc. Sum of exterior angles of an n-gon = 180n degrees.
7 Quadrilaterals A quadrilateral is any four-sided polygon but certain ones with specific restrictions are most important for us. Definition: A parallelogram is a quadrilateral whose opposite sides are parallel. A B C D A diagonal divides it into two triangles. Are the triangles congruent? Why? It follows now that the opposite sides of a parallelogram are congruent. Likewise, the opposite angles are congruent and the consecutive angles are supplementary. Converses true?
8 Theorem: The diagonals of a parallelogram bisect each other. Proof:? Question: Is the converse true? That is, if the diagonals of a quadrilateral bisect each other, must the quadrilateral be a parallelogram?
9 Definition: A parallelogram with all four sides congruent is called a rhombus. Definition: A parallelogram with all four angles congruent (each is 90 ) is called a. rectangle Definition: A rectangle with all four sides congruent is called a square.
10 Diagonals Parallelogram Rectangle Rhombus Square Bisect each other X X X X Are congruent X X Are perpendicular X X Bisect vertex angles X X Form 2 pairs of congruent triangles X X X X Form 4 congruent triangles X X
11 Areas A square unit is the surface enclosed by a square whose side measures 1 unit (like inches, feet, or some arbitrary scale.) The area of a polygon is the number of square units contained within its boundary. Area of a Rectangle: Pick one of the sides and call it the base and one of the sides perpendicular to it and call it the height. For example, if the base is 5 units and the height is 4 units, then we can fit 5x4 = 20 square units inside; so its area is 20. height = 4 units Area = 20 sq. units base = 5 units
12 Since a rectangle with a base b units long and height h units high contains bh square units, we know that The area A of a rectangle with base b a height h is A = bh Note: We sometimes use the letter A for a vertex and also for the area so make sure of the context. The area of a square with side s is A = s 2. Problem: How many square inches are there in 1 square foot? Problem: What is the area of a square with a perimeter of 36 units?
13 Area of a Parallelogram Definition: In a parallelogram, pick one of the sides and call it a base. The height is a segment from a point on the opposite side and drawn perpendicular to that base. h b Theorem: The area of a parallelogram is the base times the height; i.e., A = bh. Proof? Problem: Find the base of a parallelogram whose area is 27 and the base is three times the height.
14 Area of a Triangle In a triangle, pick one of the sides and call it the base. From the opposite vertex draw a segment perpendicular to the base; we call this the height of the triangle. h h b b Theorem: The area of a triangle is 1/2 the base times the height; A = bh/2 Proof? Problem: Find the area of a triangle whose sides are 3,4 and 5 units.
15 Area of a Trapezoid Definitions. A trapezoid is a quadrilateral with only two sides parallel; the parallel sides are called the bases and the height is a segment from one base drawn perpendicular to the other base. b h b Theorem: The area of a trapezoid is 1/2 the sum of the bases times the height; A = (b + b )h/2. Proof? Problem: What is the height of a trapezoid with bases 13 and 7 and area 40?
16 Exercises 1. What is the sum of the interior angles of an octagon? 2. What is the measure of an interior angle of a regular pentagon. 3. Prove that if the perpendiculars to two sides of a triangle from the midpoint of the third side are congruent, then the triangle is isosceles. 4. Find the area of a rectangle with base 25 and perimeter Find the area of a square whose diagonal is 50 inches. 6. Find the area of an isosceles triangle if one side is 6 and the other two sides are 5 units. 7. Area of a Kite. In the figure, AB is congruent to BC, AD is congruent to DC and BD is perpendicular to AC. Show that the area of ABCD = (1/2)(BD)(AC). B A E C D
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