1. Lesson 1-4 Angles (page 17) Essential Question How are the relationships of geometric figures used in real life situations?

2. ANGLE: a figure formed by two rays that have the same __________________. example: Name the angle as: _________________________________________________. To measure angles (in degrees in this course), use a ________________________. endpoint vertex side B C ∙ 1 ∠B∠B or ∠ ABC or ∠ CBA or ∠ 1 protractor vertex Use a protractor to measure the angle. 50º A ∙

3. Classification of Angles Acute Angle The measure is between _____ and _____ degrees. 0 90 10º 21º 35º 62º

4. Classification of Angles Right Angle The measure equals _____ degrees. 90 90º Box denotes a right angle. Rt. ∠

5. Classification of Angles Obtuse Angle The measure is between _____ and _____ degrees. 90 180 113º 142º 159º 174º

6. Classification of Angles Straight Angle The measure equals _____ degrees. 180 180º

7. Angle in a Plane (exclude straight angle) Plane is separated into 3 parts: (1) the angle itself (2) the interior of the angle (3) the exterior of the angle

8. How many angles are shown in the diagram below? _____ Name the different angles:__________________________ X..Y.Y.Z.Z O 2 3 ∠ XOZ or ∠ XOY or ∠ YOZ or ∠ 2 or ∠ 3  ∠ O 3

9. PROTRACTOR POSTULATE On line AB in a given plane, choose any point between A and B. Consider ray OA and ray OB and all rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0º to 180º in such a way that: (a) ray OA is paired with _____, and ray OB with _______. (b) If ray OP is paired with x, and ray OQ with y, then _____________ = |x – y|. POSTULATE 3 0º 180º m ∠ POQ Q.. P. P.B.B.A.A O 0º 180º yºxº

10. ANGLE ADDITION POSTULATE If point B lies in the interior of ∠ AOC, then: ______________ + ______________ = ______________. If ∠ AOC is a straight angle and B is any point not on line AC, then: ______________ + ______________ = 180º. POSTULATE 4 m ∠ AOB m ∠ BOCm ∠ AOC m ∠ AOBm ∠ BOC A. O.C.C.B.B.A.A.C.C.B.B.O.O

11. CONGRUENT ANGLES: angles that have _____________ measure. example: m ∠ A ____ m ∠ B A  A ____ ∠ B B Use a protractor to measure the angles. equal = 51º

12. ADJACENT ANGLES: two angles in a plane that have a common vertex and a common side, but no common interior points. (adj. ∠ ’s) examples: Are ∠ 1 and ∠ 2 adjacent angles? Circle Yes or No. (1) 12

13. ADJACENT ANGLES: two angles in a plane that have a common vertex and a common side, but no common interior points. (adj. ∠ ’s) examples: Are ∠ 1 and ∠ 2 adjacent angles? Circle Yes or No. (2) 1 2

14. ADJACENT ANGLES: two angles in a plane that have a common vertex and a common side, but no common interior points. (adj. ∠ ’s) examples: Are ∠ 1 and ∠ 2 adjacent angles? Circle Yes or No. (3) 1 2

15. ADJACENT ANGLES: two angles in a plane that have a common vertex and a common side, but no common interior points. (adj. ∠ ’s) examples: Are ∠ 1 and ∠ 2 adjacent angles? Circle Yes or No. (4) 1 2

16. ADJACENT ANGLES: two angles in a plane that have a common vertex and a common side, but no common interior points. (adj. ∠ ’s) examples: Are ∠ 1 and ∠ 2 adjacent angles? Circle Yes or No. (5) 1 2

17. ADJACENT ANGLES: two angles in a plane that have a common vertex and a common side, but no common interior points. (adj. ∠ ’s) examples: Are ∠ 1 and ∠ 2 adjacent angles? Circle Yes or No. (6) 1 2

18. BISECTOR of an ANGLE: the ray that divides the angle into two congruent ________________ angles. ( ∠ -bisector) example: m ∠ XOY ______ m ∠ YOZ then ∠ XOY _____ ∠ YOZ ∴ _____________ ∠ XOZ adjacent = bisects X..Z.Z O.Y.Y

19. Find all angle measures given bisects ∠ AOC. (1) m ∠ 1 = 2 x + 5 and m ∠ 2 = 3 x -12. m ∠ 1 = ________ m ∠ 2 = ________ m ∠ 3 = ________ m ∠ 1 = m ∠ 2 2x + 5 = 3x - 12 17 = x x = 17 m ∠ 1 = 2x + 5 m ∠ 1 = 2*17 + 5 m ∠ 1 = 34 + 5 m ∠ 2 = 3x - 12 m ∠ 2 = 3*17 - 12 m ∠ 2 = 51 - 12 m ∠ 1 = 39º m ∠ 1 + m ∠ 2 + m ∠ 3 = 180º 39 + 39 + m ∠ 3 = 180 m ∠ 3 = 102º m ∠ 2 = 39º 39º 102º = 180º ✔ 78 + m ∠ 3 = 180 WHY? A. B..C.C.D.D 3 2 1 O

20. Find all angle measures given bisects ∠ AOC. (2) m ∠ 1 = x + 4 and m ∠ 3 = 2 x +8. m ∠ 1 = ________ m ∠ 2 = ________ m ∠ 3 = ________ A. B..C.C.D.D 3 2 1 O m ∠ 1 + m ∠ 2 + m ∠ 3 = 180º (m ∠ 1 = m ∠ 2) x + 4 + x + 4 + 2 x + 8 = 180 4 x + 16 = 180 4 x = 164 x = 41 m ∠ 1 = x + 4 m ∠ 1 = 41 + 4 ∴ m ∠ 2 = 45º m ∠ 3 = 2 x + 8 m ∠ 3 = 2 * 41 + 8 m ∠ 1 = 45º m ∠ 3 = 82 + 8 = 90º 45º 90º = 180º ✔ WHY?

21. Assignment Written Exercises on pages 21 & 22 RECOMMENDED: 1 to 25 odd numbers REQUIRED: 24, 27 to 35 odd numbers Prepare for a quiz on Lessons 1-3 and 1-4 How are the relationships of geometric figures used in real life situations?