Introduction to Angles Latressa T R Cobb

Rays and Angles RAY: A ray, AB, starts at point A and goes on forever in one direction. A B B A C B A C ANGLE: An angle is formed by two rays with a common endpoint. VERTEX: A vertex is the point where two or more rays (or segments in a polygon) intersect. Angles can be named several ways . . . A B C A B C 1 1 means A means ABC means This 3rd method is used when more than two rays form a vertex. A B C D Which is B? This would be too confusing! Therefore we use: ABC, ABD or CBD

Rays and Angles When we want to indicate that two or more segments or angles are equal in a figure, we use an equal number of slash marks like the figures below.

Angle Vocabulary We can classify the different types of angles. Acute Angle: Any angle with a measure between (but not including) 0º and 90º. Right Angle: Any angle which measures 90º. Obtuse Angle: Any angle with a measure between (but not including) 90º and 180º. Straight Angle: Straight Angles (or lines) have a measure of 180º.

Angle Vocabulary A O B A B We will classify some of the different RELATIONSHIPS of angles. Supplementary Angles: Any pair of angles whose sum is 180º. A O B 20º 160º D 30º 150º A B LINEAR PAIR: A supplementary pair of angles that form a straight angle. Complementary Angles: Any pair of angles whose sum is 90º. 50º 40º 60º 30º

Definitions: Parallel lines: lines that are in the same plane and do not intersect (cross). Transversal: a line that intersects two or more lines in the same plane.

Definitions: Interior exterior exterior When we classify angle relationships, we need to look at the POSITION of the angles in relation to the parallel lines AND the transversal. Interior Interior means the angles are between the parallel lines. exterior Exterior means the angles are outside the parallel lines. exterior

Definitions: Two angles can be on the same side of the transversal or they can be on alternate sides of the transversal. Same side Alternate sides

Now, we put these two ideas together to classify angle relationships. Angle 4 & 5 are on alternating sides of the transversal, and are interior of the parallel lines. So, we say that angle 4 & angle 5 are Alternate Interior Angles. 2 1 4 4 3 6 5 5 8 7 **Note when we are classifying angles on parallel lines, we are dealing with PAIRS of angles classified by their positions in relation to two things: the parallel lines AND the transversal. We do NOT say that 8 is an exterior angle (this will be defined later as a single classification).

The angle relationships when parallel lines are cut by a transversal are… Alternate Interior Angles alternate sides of the transversal and interior of the parallels Same Side Interior Angles Same side of the transversal and interior of the parallels Alternate Exterior Angles alternate sides of the transversal and exterior of the parallels Same Side Exterior Angles Same side of the transversal and exterior of the parallels

There is one other important type of angle relationship on parallel lines cut by a transversal but it doesn’t quite follow the same ideas as the first four… Corresponding Angles Angles on the same side of the transversal and in the same position on the parallel lines (either the angles are both above or both below the parallel lines.) Since the five types of angles describe angle relationships, we can only classify PAIRS of angles as Alternate Interior Angles, Same Side Interior Angles, Alternate Exterior Angles, Same Side Exterior Angles, or Corresponding Angles. Pairs of angles can be classified as one of these relationships even if the lines being cut by the transversal are NOT parallel.

Find pairs of angles that are: Same side interior angles 2 2 2 (3 & 5) or (4 & 6) 1 1 1 4 4 4 3 3 3 6 6 6 5 5 5 8 8 8 Alternate interior angles 7 7 7 (3 & 6) or (4 & 5) Alternate exterior angles Corresponding angles: (1 & 8) or (2 & 7) (1 & 5) or (2 & 6) or (3 & 7) or (4 & 8) Same side exterior angles (2 & 8) or (1 & 7)

Some pairs are equal and some are supplementary. If (and only if) the lines cut by the transversal are PARALLEL, then we can make some conclusions based on the angle relationships. Some pairs are equal and some are supplementary. b a c d f e g h

So, if the lines are parallel, what can you conclude about: Alternate Interior Angles Equal Same Side Interior Angles Supplementary Alternate Exterior Angles Equal Supplementary Same Side Exterior Angles Corresponding Angles Equal

 No overlap … the common side is IN BETWEEN the two angles. Vertical Angles: The opposite angles formed by two intersecting lines. This is a relationship between PAIRS of angles…one angle cannot be vertical. Conjecture: When two lines intersect, the resulting pairs of vertical angles are ________. equal Adjacent Angles: Angles that are “next to” each other. To be adjacent angles, they MUST satisfy 3 conditions:  share a common side,  share a common vertex,  No overlap … the common side is IN BETWEEN the two angles.