2. 1Geometry Lesson: Rays, Angles Aim: Do Now: Ans. M(4, -8) Ans. 10 What are rays and angles?

3. 2Geometry Lesson: Rays, Angles Def: Rays P A A ray is a part of a line that consists of an endpoint, and all points on one side of the endpoint. Def: Opposite Rays Opposite Rays are two rays of the same line with a common endpoint and no other points in common. A B P

4. 3Geometry Lesson: Rays, Angles Def:Angle: An angle is the union of two rays having the same endpoint. side vertex Three capital letters, with vertex in the middle: Single lowercase letter or number inside the angle: B A C x Naming Angles: a) b) c) Use the name of the vertex angle if it’s the only angle at that vertex:

5. 4Geometry Lesson: Rays, Angles Angle Measure: B A C 25° Def: Congruent Anglesare angles having equal measure. Ans. The measure of an angle is the number of degrees in the angle. Q: Which of the following angles are congruent? A) 45° A C) C 45° B) B 45°

6. 5Geometry Lesson: Rays, Angles Def: Straight Angle ABO A straight angle is the union of two opposite rays. Straight angles have a measure of 180° Def: Right Angle: M Q P A right angle has a measure of 90°

7. 6Geometry Lesson: Rays, Angles Def: Acute Angle: Y XZ An acute angle has a measure greater than 0° and less than 90 °. Def: Obtuse Angle: Q X R An obtuse angle has a measure greater than 90° and less than 180°.

8. 7Geometry Lesson: Rays, Angles Def: Perpendicular Lines are two lines that intersect to form right angles. l m HK L J Right Angles: Straight Angle: Are lines m and l perpendicular? NOT UNLESS SPECIFIED BY THE GIVEN INFO OR A BOX IN THE DIAGRAM !!!!

9. 8Geometry Lesson: Rays, Angles Addition/Subtraction of Angles: If several angles share a common vertex, we can write addition and subtraction expressions using the names of the angles. R T S 20° 38° P 20° + 38° = 58° 58° - 38° = 20°

10. 9Geometry Lesson: Rays, Angles Def: Angle Bisector: Ans.: Make a ray from the vertex that divides it in half. 80° G E F 40° G E F P The bisector of an angle is a ray that divides the angle into two congruent angles. A) Or B) C)

11. 10Geometry Lesson: Rays, Angles Angle Bisector Examples Ex #1 P T M N R S a) State a pair of congruent angles. x = 4 There is no way to tell from the given info.

12. 11Geometry Lesson: Rays, Angles Ex# 2 Simple 2-Column Proof: A B C D Given Def. of Angle Bisector Def. of Congruent Angles