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1.3 a: Angles, Rays, Angle Addition, Angle Relationships

1.3 a: Angles, Rays, Angle Addition, Angle Relationships. CCSS.

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1.3 a: Angles, Rays, Angle Addition, Angle Relationships

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  1. 1.3 a: Angles, Rays, Angle Addition, Angle Relationships CCSS 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.

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

  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 Vertex- where they met ray ray

  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: • Angle ABC • Angle CBA • 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. T Vertex: Sides: Names: X XW & XT WXT TXW 4 4 5 W X Z

  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 R 30 20 Q S

  10. Find the m< CAB

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

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

  13. Types of Angle Relationships • Adjacent Angles • Vertical Angles • Linear Pairs • Supplementary Angles • Complementary Angles

  14. 1) Adjacent Angles • Adjacent Angles - Angles sharing one side that do not overlap 2 1 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 180o angle/straight angle. 2 1 3

  17. 4) Supplementary Angles • Supplementary Angles – two angles that add up to 180o (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 90o

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

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

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

  22. Solve for x and find the m<1

  23. Solve for x and find the m<1

  24. Find x and the

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