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SECTION 4.6 4.6 ANGLE MEASUREMENT CLAST OBJECTIVE " Identify relationships between angle measures Angle Measurement 169 Textbook Reference Section 8.1 Angle measurement is the measurement between two rays from a common point. Angles are usually measured in degrees. Name of Angle Basic Shape Angle Measurement Acute Less than 90 º Obtuse Greater than 90 º Right Exactly 90º Straight Exactly 180 º FACTS a) Complementary angles sum to 90 º. b) Supplementary angles sum to 180 º. c) Opposite angles formed by two intersecting lines are equal. d) Two parallel lines cut by a transversal usually create four equal obtuse angles and four equal acute angles. obtuse acute obtuse acute acute obtuse © Houghton Mifflin Company. All rights reserved. acute obtuse 170 CHAPTER 4 Geometry FACTS e) The sum of the angles in a polygon is 180( n – 2 ), where n is the number of sides of the polygon. f) Parallel lines never intersect. g) Perpendicular lines intersect at 90º . Examples a) What angle is complementary to 75 º ? Solutions 75 º + x = 90 º x = 90 º - 75 º Complementary angles sum to 90º. x = 15 º b) What angle is supplementary to 60 º ? 60 º + x = 180 º x = 180 º - 60 º Supplementary angles sum to180º. x = 120 º c) Identify two pairs of opposite angles. Angles 1 and 4 Angles 2 and 3 1 3 2 4 Note that the two angles together form a straight angle. This means the angles are supplementary and must sum to 180 º. 50 º + x = 180 º x = 180 º - 50 º = 130 º d) Find angle x. x 50º e) Give the measure of angles 1, 2, and 3. 1 2 4 6 3 5 7 60º ∠ 1 and 60 º are supplementary. ∠ 1 + 60 º = 180 º ∠ 1 = 180 º - 60 º ∠ 1 = 120 º ∠ 1 and ∠ 3 are opposite angles. Thus, ∠ 1 = ∠ 3 . Angle 3 measures 120 º . The 60 º angle and ∠ 2 are also opposite angles. ∠ 2 = 60 º . © Houghton Mifflin Company. All rights reserved. SECTION 4.6 Angle Measurement 171 Examples f) What is the measure of one angle of a regular pentagon? Solutions A pentagon has five sides. Use the sum of angles formula, 180( n – 2 ), where n = 5. 180 ( 5 – 2 ) = 180 (3) = 540 º The sum of the five angles in a pentagon is 540 º. To find the measure of just one angle, divide the total by 5. 540 ° = 108 ° 5 Each angle in a regular pentagon measures 108 º. g) If L 1 and L 2 are parallel and L 3 and L 4 are parallel, which of the following statements are true? x = 55 º because x and the 55 º angle are corresponding angles. L1 y = 180 º – 55 º = 125 º , since x and y are supplementary. L2 s t z 55º L3 r x y z = y = 125 º, since z and y are alternate interior angles L4 Remember, two parallel lines cut by a transversal usually create four equal obtuse angles and four equal acute angles. a) z = 55 º b) r = 125 º c) z = 125 º d) s = t Check Your Progress 4.6 1. Name the type of angle. a) The correct answer is C. b) c) 2. Find the complement of the following: a) 20 º b) 50 º c) 70 º 3. Find the supplement of the following: a) 40 º b) 100 º © Houghton Mifflin Company. All rights reserved. c) 135 º 172 CHAPTER 4 Geometry 4. Identify the pairs of opposite angles. t v u w 5. Find the angle measure of x. a) b) 140 º 80 º x x 6. Lines L 1 and L 2 are parallel. Find measurement of angles 1, 2, and 3. L1 1 2 3 L2 30º 7. Find x if lines L 1 and L 2 are parallel. L1 L2 3x 4x - 7 8. Find the measure of the angles if lines L 1 and L 2 are parallel. L2 L1 3x + 10 2x © Houghton Mifflin Company. All rights reserved. SECTION 4.6 Angle Measurement 173 9. Find angle x. 30 º x 10. Find the sum of the angles of a hexagon. 11. Find the sum of the angles of a rectangle. 12. Find the measure of an inside angle of a regular octagon. 13. Which statement is true for the figure shown below given that L 1 and L 2 are parallel lines? a) Since t = 80 º, s = 80 º b) Since t = 80 º, s = 100 º c) v = r d) r = 55 º q L1 p r 80 º L2 u s v t w © Houghton Mifflin Company. All rights reserved. x y 55 º 174 CHAPTER 4 Geometry See If You Remember SECTIONS 4.1 – 4.5 A 1 2 1. Round position A to the nearest inch. 3 4 5 2. Round position A to the nearest ½ inch. 3. Jack wants to put wire around a square window of side 4.5 ft. How much wire must he buy? 4. Ben is carpeting a floor 12 feet by 9 feet. How many square yards of carpet does he need? 5. A milk can is 8 inches tall and 2 inches in diameter. What is the volume of milk in a full can? 6. Find the surface area of a cube of side 4.3 ft. For Questions 7 and 8, write an appropriate measure. 7. The distance around a pool 8. The amount of water in the pool 9. Find the area of the shaded region. 4 cm 4 cm 4 cm © Houghton Mifflin Company. All rights reserved.