1. Lesson 3
Rays and Angles

2. Rays
A ray is a part of a line that starts at an endpoint & extends infinitely in one direction A ray is named by its endpoint & any other point on the ray (directional) 𝐴𝐵 & 𝐸𝐷

3. Opposite Rays
Two rays that have a common endpoint and form a line are called opposite rays 𝐵𝐴 & 𝐵𝐶 Note that 𝐴𝐵 & 𝐶𝐵 are not opposite rays

4. Angles An angle is formed by 2 rays with a common endpoint
Vertex is the common endpoint
Sides are the rays
Naming the angle
∠XYZ or ∠ZYX the vertex must be in the middle
∠Y when no adjacent angles ∠4

5. More about angles
Adjacent angles are 2 angles in the same plane that share a common vertex and a side, but share no common interior points ∠ABK & ∠KBL Overlapping angles are 2 angles in the same plane that share a common vertex and common interior points ∠ABL & ∠KBL

6. Naming Angles and Rays Name three sides 𝐽𝐼 , 𝐽𝐾 , & 𝐽𝑀
𝐽𝐼 , 𝐽𝐾 , & 𝐽𝑀
Name three angles
∠2 or ∠IJK
∠3 or ∠KJM
∠IJM
Why is ∠J a poor name?
3 angles with a common vertex

7. Measuring angles
A protractor is a tool used to measure angles in degrees One degree is of a circle Notice the protractor stops at 180° Angles are in the range of 0°<𝐴≤180°

8. Postulate 3: Protractor Postulate
Given a point O on 𝐵𝐴 , consider rays 𝑂𝐵 & 𝑂𝐴 , as well as all the other rays that can be drawn with O as the endpoint, on one side of 𝐵𝐴 . These rays can be paired with the real numbers from 0 to 180 such that:
𝑂𝐵 is paired to 0, and 𝑂𝐴 is paired with 180.
If 𝑂𝐶 is paired with a number 𝑐 and 𝑂𝐷 is paired with a number 𝑑, then 𝑚∠𝐷𝑂𝐶= 𝑑−𝑐
𝑚∠𝐷𝑂𝐶= 65−135 𝑚∠𝐷𝑂𝐶= −70 𝑚∠𝐷𝑂𝐶=70°

9. Classifying Angles
Acute– 0°<𝑎<90° Right– equal to 90° Obtuse– 90°<𝑎<180° Straight– equal to 180° Rays that form a straight angle are also called opposite rays What other figure could you call a straight angle? Line

10. Postulate 4: Angle Addition Postulate
If point D is in the interior of ∠ABC, then
m∠ABD + m∠DBC = m∠ABC
If m∠ABD = 45° and m∠ABC = 97°, then find the m∠DBC
45 + m∠DBC = 97
m∠DBC = 52°

11. More on angles
To bisect a figure is to divide it into 2 ≅ parts An angle bisector is a ray that divides an angle into 2 congruent angles Congruent angles have the same measure Congruent angles can be shown using arc marks

12. Find the m∠ABC & m∠KBL if:
𝐵𝐾 bisects ∠ABD, 𝐵𝐿 bisects ∠DBC, m∠ABK=22° & m∠CBD=50° By the definition of angle bisector ∠ABK≅∠KBD & ∠DBL≅∠LBC

13. Find the m∠ABC & m∠KBL if:
𝐵𝐾 bisects ∠ABD, 𝐵𝐿 bisects ∠DBC, m∠ABK=22° & m∠CBD=50°
By the definition of angle bisector ∠ABK≅∠KBD & ∠DBL≅∠LBC
m∠KBD=22°, m∠DBL=25°

14. Find the m∠ABC & m∠KBL if:
m∠ABC = m∠ABD + m∠DBC
m∠ABC =
m∠ABC = 94°
m∠KBL = m∠KBD + m∠DBL
m∠KBL =
m∠KBL = 47°

15. Questions/Review Angle addition can be used for multiple angles
m∠ABC =m∠1+ m∠2 + m∠3 + m∠4
Just be careful to make sure your angles are adjacent angles and not overlapping angles