NAME DATE For use with pages A polygon is convex if no line that contains a side of the polygon passes ,through the interior of the polygon. A polygon that is not convex is called concave. A polygon is equilateral if all of its sides are congruent. A polygon is equiangular if all of its interior angles are congruent. A polygon is regular if it is both equilateral and equiangular. Decide whether the polygon is convex or ,concave. None of the extended sides pass through the interior. So, the polygon is convex. At least one extended side passes through the interior. So, the polygon is concave. Copyright © McDougal Littell Inc. Chapter 8 Resource Book All rights reserved. LESSON NAME DATE [] For use with pages Decide whether the polygon is regular. Explain your answer. a. The polygon is equilateral because all of the sides are congruent. The polygon is not equiangular because not all of the angles are congruent. So, the polygon i~s not regular. b. The polygon is equilateral because all of the sides are congruent. The polygon is equiangular because all of the angles are congruent. So, the polygon is regular. The polygon is regular. Find the value of x. B__C Because the polygon is regular, all of its sides are congruent. +1 AB = DE The sides of the polygon are congruent. E 25 = 6x+ 1 Substitute 25 for AB and 6x + 1 for DE. 24 = 6x Subtract 1 from each side. 4--X Divide each side by 6. 10. 15 Copyright © McDougal Littell Inc. All rights reserved. Chapter 8 Resource Book NAME DATE For use wi~h pa~es Theorem 8.1 Polygon Interior Angles Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n - 2) ¯ 180°. Theorem 8.2 Polygon Exterior Angles Theorem ¯ The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°. ~ Find the sum of the measures of the interior angles of the polygon. The polygon has six sides (hexagon). Use the Polygon Interior Angles Theorem and substitute 6 forn. (n - 2). 180° = (6 - 2) ¯ 180° Substitute 6 for n. = 4 ¯ 180° Simplify. = 720° Multiply. ............................................................................................. Find the sum of the measures of the interior angles of the polygon. 1. quadrilateral 2, pentagon 3o octagon Find the measure of Z A in the diagram. A B ~135° 83°~, D C Geometry Copyright © McD0ugal Littell Inc. Chapter 8 Resource Book All rights reserved. NAME DATE Far use with pages ~,16-423 The polygon has five sides, so the sum of the measures of the interior angles is (n - 2). 180° = (5 - 2). 180° = 3. 180° = 540°: Add the measures of the interior angles and set the sum equal to 540°. 90° + 135° + 83° + 96° + mZA = 540° The sum is 540°. 404° + mZA = 540° Simplify. mZA = 136° Subtract 404° from each side. Exercises for Example 2 F~d the ~easure ~f Z A. 4. Bj 5. E F C A D B c D D c E Find the value of x. Using the Polygon Exterior Angles Theorem, set the sum of the exterior angles equal to 360°. 130°+x°+100°+30°+x°=360° Polygon Exterior Angles Theorem 260 + 2x = 360 Add like terms. 2x = 100 Subtract 260 from each side. x = 50 Divide each side by 2. Answer: The value of x is 50. , Exercises fer Example 3 Find the va~ue of x. <70°\ 147°" Copyright © McD0ugal Littell Inc. All rights reserved. Chapter 8 Resource Book NAME DATE For use with pages 424-429 Find the area of squares and rectangles. The amount of surface covered by a figure is its area. Area of a Square: Area = (side)2 Area of a Rectangle: Area = (base)(height) Find the Area of a Square and the Area of a RectanOle a. Find the area of the square. b. Find the area of the rectangle. 3 cm 22 cm a. Use the formula for the area of a square and substitute 6 for s, A = s2 Formula for the area of a square = 62 Substitute 6 for s. = 36 Simplify. Answer: The area of the square is 36 square inches. b. Use the formula for the area of a rectangle and substitute 22 for b and 3 for h. A = bh Formula for the area of a rectangle , = 22 ¯ 3 Substitute 22 for b and 3 for h. = 66 Simplify. Answer: The area of the rectangle is 66 square centimeters. Exercises for Example I , . ................................. Find the area of the square or rectangle. 1. rectangle 2. square 3. rectangle 6 cm 4in. lOft 15 crn 8ft Copyright © McD0ugal Littell Inc. Chapter 8 Resource Book All rights reserved. I NAME DATE COi~Tli~! U ~= D For use with pages a,2o,-42g The rectangle has an area of 132 square feet. Find its base. llft A = bh Formula for the area of a rectangle 132 = b ¯ 11 Substitute 132 for A and 11 for h. 12 = b Divide each side by 11. Answer: The base of the rectangle is 12 feet. ................... ¯ ........................................................................... A gives the area of the rectangJe. Find the missing side Jength. 6. h h 20 cm 9ft A = 12 in.2 A = 81 ft2 A = 140 cm2 Find the area of the polygon made up of rectangles, t~ 10 in. 4in. Add the areas of the two rectangles. Notice that the 1 in. base and the height of rectangle A are given in the 4in. diagram. The base of rectangle B is 10 inches minus 4 inches. The height of rectangle B is 4 inches minus 1 inch. Area = Area of A + Area of B =4.4+(10-4).(4- 1) =4"4+6.3 = 16 + 18 = 34 in.2 Exercises for Example 3 2 cm 5cm ~ 15 in. ~1 I 11 cm I Copyright © McDougal Littell Inc. All rights reserved. Chapter 8 Resource Book NAME DATE For use with pages ~0-o,38 The height ef a .triangle is the perpendicular segment from a vertex to the line containing the opposite side, called the base of the triangle. Area of a ~iangle: ~ea = ~(base)(height) Theorem 803 Areas of S~mflar P~lyg~ns If two polygons simil~with a scale factor of ~, then the ratio of ~2 their ~eas is -- b~" Find the Area of a Rioht Triangle Find the area of the right triangle. Use the formula for the area of a triangle and 15 yd substitute 15 for b and 8 for h. A = ~bh Formula for the area of a triangle ’ = ~(15)(8) Substitute 15 for b and 8 for h. = 60 Simplify. Answer." The right triangle has an area of 60 square yards. Find the area of the right triangle. 1. 2. 3. 5 in. 8cm 12 in. 18 cm 4ft Find the area of the triangle. A = ½bh Formula for the area of a triangle 8 cm = ½(8)(4) Substitute 8 for b and 4 for h. = 16 Simplify. Answer: The triangle has an area of 16 square centimeters. Copyright © McD0ugal Littell Inc. Chapter 8 Resource Book All rights reserved. NAME DATE For use with pages ~0-~38 Exercises fer Example 2 ¸4. 14ft = 20 cm 9in. Find the base of the triangle, given that its area is 42 square feet. A = ½bh Formula for the area of a triangle 1 42 = ~b. 6 Substitute 42 for A and 6 for h. 84=b.6 Multiply each side by 2. 14 = b Divide each side by 6. Answer." The triangle has a base of 14 feet. Exercises fer Exampie 3 , A gives the area of the triangle. Find the missing measure. 7. A= 15in.2 8. A= 126cm2 9. A=6ft2 6ft 10 in. Areas of Similar Tdang~es /~ABC ~/~DEF. Find the scale factor of /~DEF to/~ABC. Then find the ratio of their areas. 5 C D ~;~TB~ 10 F The scale factor of ADEF to/~ABC is 4 _ 2 Then by Theorem 8.3, the ratio of the 2 1" 22 -J-’4 You can verify this by finding the~ ~eas. ~eas of ~DEF to ~ABC is ~ = Exercise for E~amp~e 4 ~ ~. ~ABC ~ GD£F. FJnd th~ sca]~ ~acto~ o~ B E ~DEF to ~ABC. Then find the ~ado o~ the~ ~eas. F 10 C Copyright © McDouga[ Littell Inc, All rights reserved. Chapter 8 Resource Book NAME DATE For use with pages 439-4a,5 Find the area of parallelograms. Either pair of parallel sides of a parallelogram are called the bases la parallelogram. The shortest distance between the bases of a parallelogram is called the height ot" a parallelogram. Area of a Parallelogram: Area = (base)(height) i 1 Area of a Rhombus: Area = =-(productz- of diagonals) Find the Area Find the area of the parallelogram. Use the formula for the area of a parallelogram and 8 yd substitute 8 for b and 6 for h. A = bh Formula for the area of a parallelogram = (8)(6) Substitute 8 for b and 6 for h. = 48 Multiply. Answer: The parallelogram has an area of 48 square yards. ............................................................................................. Find the area of the para~e~ogramo :2. 12 cm 17 cm 5m . 4. 12 yd = 10 yd 14ft Copyright © McDougal Littell Inc. All rights reserved. NAME DATE For use with pages Find the base of the parallelogram given that its area is 105 square inches. Use the formula for the area of a parallelogram and substitute 105 for A and 7 for h. A = bh Formula for the’ area of a parallelogram 105 = b ¯ 7 Substitute 105 for A and 7 for h.