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Copyright © 2005 Pearson Education, Inc. Introduction to Trigonometry Angle Relationships and Similar Triangles Copyright © 2005 Pearson Education, Inc. Basic Terms continued Angle-formed by rotating a ray around its endpoint. The ray in its initial position is called the initial side of the angle. The ray in its location after the rotation is the terminal side of the angle. Copyright © 2005 Pearson Education, Inc. Slide 1-3 Basic Terms continued Positive angle: The rotation of the terminal side of an angle counterclockwise. Copyright © 2005 Pearson Education, Inc. Negative angle: The rotation of the terminal side is clockwise. Slide 1-4 Standard Position An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis. Angles in standard position having their terminal sides along the x-axis or y-axis, such as angles with measures 90, 180, 270, and so on, are called quadrantal angles. Copyright © 2005 Pearson Education, Inc. Slide 1-5 Coterminal Angles A complete rotation of a ray results in an angle measuring 360. By continuing the rotation, angles of measure larger than 360 can be produced. Such angles are called coterminal angles. Copyright © 2005 Pearson Education, Inc. Slide 1-6 Angles and Relationships q m n Name Angles Rule Alternate interior angles 4 and 5 3 and 6 Angles measures are equal. Alternate exterior angles 1 and 8 2 and 7 Angle measures are equal. Interior angles on the same side of the transversal 4 and 6 3 and 5 Angle measures add to 180. Corresponding angles 2 & 6, 1 & 5, 3 & 7, 4 & 8 Angle measures are equal. Copyright © 2005 Pearson Education, Inc. Slide 1-7 Conditions for Similar Triangles Corresponding angles must have the same measure. Corresponding sides must be proportional. (That is, their ratios must be equal.) Copyright © 2005 Pearson Education, Inc. Slide 1-8 Example: Finding Angle Measures Triangles ABC and DEF are similar. Find the measures of angles D and E. D Since the triangles are similar, corresponding angles have the same measure. Angle D corresponds to angle A which = 35 A 112 35 F C 112 33 Copyright © 2005 Pearson Education, Inc. E Angle E corresponds to angle B which = 33 B Slide 1-9 Example: Finding Side Lengths Triangles ABC and DEF are similar. Find the lengths of the unknown sides in triangle DEF. 32 64 16 x 32 x 1024 x 32 D A 16 112 35 64 F 32 C 112 33 48 Copyright © 2005 Pearson Education, Inc. To find side DE. B E To find side FE. 32 48 16 x 32 x 768 x 24 Slide 1-10 Example: Complementary Angles Find the measure of each angle. Since the two angles form a right angle, they are complementary angles. Thus, k 20 k 16 90 k +20 k 16 2k 4 90 2 k 86 k 43 Copyright © 2005 Pearson Education, Inc. The two angles have measures of 43 + 20 = 63 and 43 16 = 27 Slide 1-11 Example: Coterminal Angles Find the angles of smallest possible positive measure coterminal with each angle. a) 1115 b) 187 Add or subtract 360 as may times as needed to obtain an angle with measure greater than 0 but less than 360. o o o a) 1115 3(360 ) 35 b) 187 + 360 = 173 Copyright © 2005 Pearson Education, Inc. Slide 1-12 Example: Finding Angle Measures Find the measure of each marked angle, given that lines m and n are parallel. (6x + 4) (10x 80) m n The marked angles are alternate exterior angles, which are equal. Copyright © 2005 Pearson Education, Inc. 6 x 4 10 x 80 84 4 x 21 x One angle has measure 6x + 4 = 6(21) + 4 = 130 and the other has measure 10x 80 = 10(21) 80 = 130 Slide 1-13