Copyright © 2005 Pearson Education, Inc.

Introduction to Trigonometry Angle Relationships and Similar Triangles

Copyright © 2005 Pearson Education, Inc. Slide 1-3 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 1-4 Basic Terms continued Positive angle: The rotation of the terminal side of an angle counterclockwise. Negative angle: The rotation of the terminal side is clockwise.

Copyright © 2005 Pearson Education, Inc. Slide 1-5 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 1-6 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 1-7 Angles and Relationships m n q Angle measures are equal.2 & 6, 1 & 5, 3 & 7, 4 & 8 Corresponding angles Angle measures add to 180 . 4 and 6 3 and 5 Interior angles on the same side of the transversal Angle measures are equal.1 and 8 2 and 7 Alternate exterior angles Angles measures are equal.4 and 5 3 and 6 Alternate interior angles RuleAnglesName

Copyright © 2005 Pearson Education, Inc. Slide 1-8 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 1-9 Example: Finding Angle Measures Triangles ABC and DEF are similar. Find the measures of angles D and E. Since the triangles are similar, corresponding angles have the same measure. Angle D corresponds to angle A which = 35  Angle E corresponds to angle B which = 33  A CB F E D 35  112  33  112 

Copyright © 2005 Pearson Education, Inc. Slide 1-10 Example: Finding Side Lengths Triangles ABC and DEF are similar. Find the lengths of the unknown sides in triangle DEF. To find side DE. To find side FE. A CB F E D 35  112  33  112 

Copyright © 2005 Pearson Education, Inc. Slide 1-11 Example: Complementary Angles Find the measure of each angle. Since the two angles form a right angle, they are complementary angles. Thus, k  16 k +20 The two angles have measures of = 63  and 43  16 = 27 

Copyright © 2005 Pearson Education, Inc. Slide 1-12 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. a) b)  187   = 173  (360 )   35   

Copyright © 2005 Pearson Education, Inc. Slide 1-13 Example: Finding Angle Measures Find the measure of each marked angle, given that lines m and n are parallel. The marked angles are alternate exterior angles, which are equal. One angle has measure 6x + 4 = 6(21) + 4 = 130  and the other has measure 10x  80 = 10(21)  80 = 130  m n (10x  80)  (6x + 4) 