1. 1 1-6 Measuring Angles Objectives: Define and name angles, sides, and rays Use the Protractor Postulate for measuring angles Classify angles as acute, right, obtuse, or straight Use the Angle Addition Postulate Define vertical angles, adjacent angles, complementary angles, and supplementary angles

2. 2 Angle, Sides, Vertex An angle is a figure formed by two rays that have a common endpoint. The rays are the sides of the angle. (rays BT and BQ) The common endpoint is called the vertex of the angle (point B). When naming an angle with 3 letters, the vertex must be the middle letter. B Q T 1 Names: QBT 1 B TBQ

3. 3 Naming angles What are two other names for ∠ 1? ∠ XWY, ∠ YWX Is ∠ W a good name for ∠ 1? No, it would not be clear which angle ∠ W would be referring to. W Y X Z 1 2

4. 4 An angle separates a plane into three parts: 1)the ______, which is the set of points between the sides of the angle 2) the ______, which is the set of points outside the angle 3) the _________ interior exterior angle itself exterior interior W Y Z A B In the figure shown, point B and all other points in the blue region are in the interior of the angle. Point A and all other points in the green region are in the exterior of the angle. Points Y, W, and Z are on the angle. Interior and exterior

5. 5 Measuring Angles We measure an angle using a protractor. –Determine the amount of rotation between the two sides of an angle. For every angle, there is a unique positive number between 0 and 180 called the degree measure of the angle. Special angles: –0°, 90°, 180°, 360° Simulation or hands-on for measuring angles: http://www.mathcasts.org/gg/student/angles/angles/angle_meas3.html

6. 6 Use a protractor to draw an angle having a measure of 135. 1) Draw AB 2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray. 3) Locate and draw point C at the mark labeled 135. Draw AC. C A B Drawing an Angle with a Protractor

7. 7 Classifying Angles A right angle m A = 90 acute angle 0 < m A < 90 A obtuse angle 90 < m A < 180 A A straight angle m A = 180

8. 8 Congruent Angles Angles with the same measure m  1 = m  2 (the measure of angle 1 equals the measure of angle 2) Ð1 ≅  2 (Angle 1 is congruent to angle 2) (May also be indicated by arc on both angles) 1 2

9. 9 1) Draw an acute, an obtuse, or a right angle. Label the angle AOC. A C O 2) Draw and label a point B in the interior of the angle. Then draw OB. B 3)For each angle, find m  AOC m  COB m  AOB. 30° 45° 75° Hands-on Measurement of Angles

10. 10 Angle Addition Postulate For any angle AOC, if B is in the interior of  AOC, then m  AOB + m  BOC = m  AOC. A C O B 30° 45° 75°

11. 11 p. 38 TxtBk Ex. 3 What is m ∠TSW if m ∠RST = 50 and m ∠RSW = 125 ? R W T S m ∠ RST + m ∠ TSW = m ∠ RSW 50 + m ∠ TSW = 125 m ∠ TSW = 125 – 50 = 75 125° 50°

12. 12 Identifying Angle Pairs – Adjacent Angles Adjacent (next to, joining) angles are angles that: M J N R 1 2  1 and  2 are adjacent with the same vertex R and common side A) share a common side B) have the same vertex C) have no interior points in common D) are coplanar

13. 13 Determine whether  1 and  2 are adjacent angles. No. They have a common vertex B, but _____________ no common side 1 2 B 1 2 G Yes. They have the same vertex G and a common side with no interior points in common. N 1 2 J L No. They do not have a common vertex nor ____________ a common side The side of  1 is ____ The side of  2 is ____ Identifying Angle Pairs: Adjacent Angles

14. 14 Identifying Angle Pairs: Vertical Angles Vertical Angles Two angles are vertical if and only if they are two nonadjacent angles formed by a pair of intersecting lines. 1 2 3 4 Vertical angles:  1 and  3  2 and  4 A ngles V ertical

15. 15 Identifying Angle Pairs: Complementary Angles Two angles are complementary if and only if the sum of their degree measures is 90. Each angle is a complement of the other. (Angle B is the complement of angle E) 30° A B C 60° D E F

16. 16 15° H 75° I Some examples of complementary angles are shown below. m  H + m  I = 90 m  PHQ + m  QHS = 90 Remember: C omplementary angles can form a C orner (which measures 90°). 50° H 40° Q P S 30° 60° T U V W Z m  TZU + m  VZW = 90 Complementary Angles: Examples

17. 17 Identifying Angle Pairs: Supplementary Angles Two angles are supplementary if and only if the sum of their degree measures is 180. 50° A B C 130° D E F m  B+ m  E = 50 + 130 = 180

18. 18 105° H 75° I Some examples of supplementary angles are shown below. m  H + m  I = 180 m  PHQ + m  QHS = 180 Remember: S upplementary angles can form a linear pair or S traight line (which measures 180°) 50° H 130° Q P S m  TZU + m  UZV = 180 60° 120° T U V W Z 60° and m  TZU + m  VZW = 180 Supplementary Angles: Examples

19. 19 Linear Pair A pair of adjacent angles whose noncommon sides that form opposite rays Hands-On: –On your paper, draw a linear pair –Measure each of the two angles and add the measures Simulation: http://www.geogebra.org/en/upload/files/english/Barbara_Perez/Linear_Angles.html 50° H 130° Q P S