1.3 a: Angles, Rays, Angle Addition, Angle Relationships G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent;

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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

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

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

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 W X Z T Vertex: Sides: Names: X 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 Q S R 30 20

Find the m< CAB

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

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

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

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

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?

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

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

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

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?

Solve for x and find the m<1

Find x and the