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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. Rays • A ray extends forever in one direction • Has one endpoint • The endpoint is used first when naming the ray B ray RB R T ray WT W 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 Angles (cont.) • Measured in degrees • Congruent angles have the same measure 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 Ex. of naming an angle • Name the vertex and sides of 4, and give all possible names for 4. T Vertex: X 4 W X 5 Sides: Z Names: XW & XT WXT TXW 4 Name the angle shown as 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 Angle Addition Postulate • If R is in the interior of PQS, then m PQR + m RQS = m PQS. P R Q 30 20 S Find the m< CAB Example of Angle Addition Postulate 100 Ans: x+40 + 3x-20 = 8x-60 60 40 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 Find the m< BYZ -2a+48 4a+9 4a+9 Types of Angle Relationships 1. 2. 3. 4. 5. Adjacent Angles Vertical Angles Linear Pairs Supplementary Angles Complementary Angles 1) Adjacent Angles • Adjacent Angles - Angles sharing one side that do not overlap 2 1 3 2)Vertical Angles • Vertical Angles - 2 non-adjacent angles formed by 2 intersecting lines (across from each other). They are CONGRUENT !! 1 2 3) Linear Pair • Linear Pairs – adjacent angles that form a straight line. Create a 180o angle/straight angle. 2 1 3 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? 5) Complementary Angles • Complementary Angles – the sum of the 2 angles is 90o Angle Bisector • A ray that divides an angle into 2 congruent adjacent angles. A D B C BD is an angle bisector of <ABC. YB bisects <XYZ 40 What is the m<BYZ ? Last example: Solve for x. BD bisects ABC A D x+40=3x-20 40=2x-20 B C 60=2x 30=x Why wouldn’t the Angle Addition Postulate help us solve this initially? Solve for x and find the m<1 Solve for x and find the m<1 Find x and the DBC