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Understanding Angles and Their Relationships in Geometry

This comprehensive guide explores fundamental concepts in geometry, focusing on angles, rays, and various angle relationships. Learn about angle measurement, ray properties, and theorems including congruence of vertical angles and the relationships created by transversals intersecting parallel lines. Detailed explanations cover naming angles, the Angle Addition Postulate, and classifications of angles (acute, right, obtuse). Dive into adjacent angles, vertical angles, linear pairs, as well as supplementary and complementary angles. This resource is essential for mastering geometry concepts in accordance with the Common Core standards.

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Understanding Angles and Their Relationships in Geometry

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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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