1. Aim: Do Now: What are rays and angles? Ans. M(4, -8) Ans. 10
Geometry Lesson: Rays, Angles

2. Def: Rays Def: Opposite Rays
A ray is a part of a line that consists of an endpoint, and all points on one side of the endpoint. P A
Def: Opposite Rays
Opposite Rays are two rays of the same line with a common endpoint and no other points in common. A B P
Geometry Lesson: Rays, Angles

3. Def:Angle: An angle is the union of two rays having the same endpoint.
side B A C x
vertex
Naming Angles: a) b) c)
Three capital letters, with vertex in the middle:
Single lowercase letter or number inside the angle:
Use the name of the vertex angle if it’s the only angle at that vertex:
Geometry Lesson: Rays, Angles

4. The measure of an angle is the number of degrees in the angle.
Angle Measure:
The measure of an angle is the number of degrees in the angle. B A C
25°
Def: Congruent Angles
are angles having equal measure.
Q: Which of the following angles are congruent? A)
45° A C) C B) B
Ans.
Geometry Lesson: Rays, Angles

5. Def: Straight Angle
A straight angle is the union of two opposite rays.
Straight angles have a measure of 180° A B O
Def: Right Angle: M Q P
A right angle has a measure of 90°
Geometry Lesson: Rays, Angles

6. An acute angle has a measure greater than 0° and less than 90 °.
Def: Acute Angle:
An acute angle has a measure greater than 0° and less than 90 °. Y X Z
Def: Obtuse Angle: Q X R
An obtuse angle has a measure greater than 90° and less than 180°.
Geometry Lesson: Rays, Angles

7. Def: Perpendicular Lines
are two lines that intersect to form right angles. l m
Are lines m and l perpendicular?
NOT UNLESS SPECIFIED BY THE GIVEN INFO OR A BOX IN THE DIAGRAM !!!! H K L J
Right Angles:
Straight Angle:
Geometry Lesson: Rays, Angles

8. Addition/Subtraction of Angles:
If several angles share a common vertex, we can write addition and subtraction expressions using the names of the angles. R T S
20°
38° P
20° ° = 58°
58° ° = 20°
Geometry Lesson: Rays, Angles

9. Ans.: Make a ray from the vertex that divides it in half.
Def: Angle Bisector:
The bisector of an angle is a ray that divides the angle into two congruent angles.
40° G E F P
80° G E F
Ans.: Make a ray from the vertex that divides it in half. A) Or B) C)
Geometry Lesson: Rays, Angles

10. Angle Bisector Examples
a) State a pair of congruent angles.
There is no way to tell from the given info.
x = 4
Geometry Lesson: Rays, Angles

11. Ex# 2 Simple 2-Column Proof:A B C D
Given
Def. of Angle Bisector
Def. of Congruent Angles
Geometry Lesson: Rays, Angles