You can use the postulates below to prove several theorems.

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1 Using Area Formulas You can use the postulates below to prove several theorems. AREA POSTULATES Postulate Area of a Square Postulate The area of a square is the square of the length of its side, or s. Postulate 3 Area Congruence Postulate If two polygons are congruent, then they have the same area. Postulate 4 Area Addition Postulate The area of a region is the sum of the areas of its nonoverlapping parts.

2 Using Area Formulas Area Addition Postulate The area of a region is the sum of the areas of its nonoverlapping parts. You can justify the Area Addition Postulate as follows.

3 What is the total shaded area? Now what is the total shaded area?

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5 Using Area Formulas AREA THEOREMS Theorem 6.0 Area of a Rectangle The area of a rectangle is the product of its base and its height. b h Theorem 6. Area of a Parallelogram The area of a parallelogram is the product of a base and its corresponding height. b h Theorem 6. Area of a Triangle The area of a triangle is one half the product of a base and its corresponding height. XXX b h XXX

6 Using Area Formulas You can justify the area formulas for triangles and parallelograms as follows. The area of a parallelogram is the area of a rectangle with the same base and height. The area of a triangle is half the area of a parallelogram with the same base and height.

7 Using the Area Theorems Find the area of ABCD. SOLUTION Method Use AB as the base. So, b = 6 and h = 9. Area = b h = 6(9) = 44 square units Method Use AD as the base. So, b = and h =. Area = b h = () = 44 square units Notice that you get the same area with either base.

8 Finding the Height of a Triangle Rewrite the formula for the area of a triangle in terms of h. SOLUTION Rewrite the area formula so h is alone on one side of the equation. b h Formula for the area of a triangle b h Multiply both sides by. A b = h Divide both sides by b.

9 Finding the Height of a Triangle A triangle has an area of 5 square feet and a base of 3 feet. Are all triangles with these dimensions congruent? SOLUTION Using the formula for the height of a triangle, the height is h = (5) 3 = 8 feet. There are many triangles with these dimensions. Some are shown below.

10 Areas of Trapezoids, Kites, and Rhombuses THEOREMS Theorem 6.3 Area of a Trapezoid The area of a trapezoid is one half the product of the height and the sum of the bases. h (b + b ) Theorem 6.4 Area of a Kite The area of a kite is one half the product of the lengths of its diagonals. d d Theorem 6.5 Area of a Rhombus The area of a rhombus is equal to one half the product of the lengths of the diagonals. d d

11 Areas of Trapezoids, Kites, and Rhombuses You may find it easier to remember the Area of a Trapezoid by using the midsegment. Area of a Trapezoid = Length of Midsegment Height

12 Finding the Area of a Trapezoid Find the area of trapezoid W X Y Z. SOLUTION The height of W X Y Z is h = 5 = 4. Find the lengths of the bases. b = Y Z = 5 = 3 b = X W = 8 = 7 Substitute 4 for h, 3 for b, and 7 for b to find the area of the trapezoid. h ( b + b ) h 4 ( b b ) = 0 The area of trapezoid WXYZ is 0 square units. Formula for the area of a trapezoid Substitute. Simplify.

13 Areas of Trapezoids, Kites, and Rhombuses The diagram justifies the formulas for the areas of kites and rhombuses. d d The diagram shows that the area of a kite is half the area of the rectangle whose length and width are the diagonals of the kite. The same is true for a rhombus.

14 Finding the Area of a Rhombus Use the information given in the diagram to find the area of rhombus ABCD. SOLUTION Method Use the formula for the area of a rhombus. d = BD = 30 and d = AC = 40 d d (30)(40) = 600 square units

15 Finding the Area of a Rhombus Use the information given in the diagram to find the area of rhombus ABCD. SOLUTION Method Use the formula for the area of a rhombus. d = BD = 30 and d = AC = 40 d d (30)(40) = 600 square units Method Use the formula for the area of a parallelogram. b = 5 and h = 4. b h = 5(4) = 600 square units

16 Finding Areas ROOF Find the area of the roof. G, H, and K are trapezoids and J is a triangle. The hidden back and left sides of the roof are the same as the front and right sides. SOLUTION Area of J = (0)(9) = 90 ft = (5)(0 + 30) = 375 ft Area of H = (5)(4 + 50) = 690 ft Area of K = 43 ft Area of G = ()(30 + 4) The roof has two congruent faces of each type. Total Area = ( ) = 374 The total area of the roof is 374 square feet.