1. 1.3 a: Angles, Rays, Angle Addition, Angle Relationships G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. CCSS

2. Rays A ray extends forever in one direction Has one endpoint The endpoint is used first when naming the ray R Bray RB T W ray WT R B R B R B R

3. Angles Angles are formed by 2 non-collinear rays The sides of the angle are the two rays The vertex is where the two rays meet ray Vertex- where they met

4. Angles (cont.) Measured in degrees Congruent angles have the same measure

5. Naming an Angle You can name an angle by specifying three points: two on the rays and one at the vertex. The angle below may be specified as angle ABC or ABC. The vertex point is always given in the middle. Named: 1)Angle ABC 2)Angle CBA 3)Angle B * *you can only use the vertex if there is ONE angle Vertex

6. Ex. of naming an angle Name the vertex and sides of 4, and give all possible names for 4. 4 5 W X Z T Vertex: Sides: Names: X XW & XT WXT TXW 4

7. Name the angle shown as

8. Angles can be classified by their measures Right Angles – 90 degrees Acute Angles – less than 90 degrees Obtuse Angles – more than 90, less than 180

9. Angle Addition Postulate If R is in the interior of PQS, then m PQR + m RQS = m PQS. P Q S R 30 20

10. Find the m< CAB

11. Example of Angle Addition Postulate Ans: x+40 + 3x-20 = 8x-60 4x + 20 = 8x – 60 80 = 4x 20 = x Angle PRQ = 20+40 = 60 Angle QRS = 3(20) -20 = 40 Angle PRS = 8 (20)-60 = 100 60 40 100

12. 4a+9 -2a+48 Find the m< BYZ

13. Types of Angle Relationships 1.Adjacent Angles 2.Vertical Angles 3.Linear Pairs 4.Supplementary Angles 5.Complementary Angles

14. 1) Adjacent Angles Adjacent Angles - Angles sharing one side that do not overlap 1 2 3

15. 2)Vertical Angles Vertical Angles - 2 non-adjacent angles formed by 2 intersecting lines ( across from each other ). They are CONGRUENT !! 1 2

16. 3) Linear Pair Linear Pairs – adjacent angles that form a straight line. Create a 180 o angle/straight angle. 1 2 3

17. 4) Supplementary Angles Supplementary Angles – two angles that add up to 180 o (the sum of the 2 angles is 180) Are they different from linear pairs?

18. 5) Complementary Angles Complementary Angles – the sum of the 2 angles is 90 o

20. Angle Bisector A ray that divides an angle into 2 congruent adjacent angles. BD is an angle bisector of <ABC. B A C D

21. YB bisects <XYZ 40 What is the m<BYZ ?

22. Last example: Solve for x. x+40 o 3x-20 o x+40=3x-20 40=2x-20 60=2x 30=x B C D BD bisects ABC A Why wouldn’t the Angle Addition Postulate help us solve this initially?

23. Solve for x and find the m<1

25. Find x and the