1. 1-4 Rays, Angles and Angle Measures
Honors Geometry

2. Standard/Objectives:
Objective 1: Students will understand geometric concepts and applications
Objective 2: Students will use visualization, spatial reasoning, and geometric modeling to solve problems.
Objective 3: Students will use angle postulates to Classify angles as acute, right, obtuse, or straight.

3. Ray
Definition: An infinite set of points that has a definite starting point called the “ENDPOINT” and extends to infinity in ONE direction.
SYMBOL: An arrow pointing to the right

4. Rays: Cont.
Naming: named by using the endpoint first and any point on the ray.
ray AB or AB A B

5. Example 1: Name ALL the rays & all the DIFFERENT rays:
All = every ray every way
Different = each ray one time A B C D E

6. Collinear Rays
Definition: When the points of two rays ALL lie on the same line.
Name ALL the rays:
AD, AR, DR, RD, RA & DA
Name all the DIFFERENT rays:
AR, DR, RA & DA R D A

7. Opposite Rays
Definition: Two rays that are collinear and have only their endpoints in common.
rays BA & BC or BA & BC
ALSO CALLED STRAIGHT ANGLES A B C

8. Angle
Definition: a figure consisting of two of two NON-COLLINEAR rays with a common endpoint.
Sides: the two rays with the common endpoint that form the angle. BA & BC
Vertex: the common endpoint that makes the angle. point B

9. Angles: cont.
Naming: ∠
Three points name an angle (the vertex & one point from each ray), the vertex must be the middle point. ∠CBA or ∠ABC
A number or lowercase letter. ∠1 or ∠a
The vertex. ∠B

10. Angles: cont.
NOTE: if more than one angle share the same vertex then you CAN’T use the vertex to name the angle.

11. EXAMPLE 1
Name angles
Name the three angles in the diagram.
∠ WXY, or ∠YXW
∠YXZ, or ∠ ZXY
∠ WXZ, or ∠ ZXW
You should not name any of these angles ∠ X because all three angles have point X as their vertex.

12. Angles: cont. An angle separates a plane into three parts.
INTERIOR: point F is on the interior of ∠ V
EXTERIOR: point S is on the exterior of ∠V
ON: point H is on ∠V H S F

13. Classifying Angles
Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0° and less than or equal to 180°.

14. 1. Name all the angles in the diagram. Which angle is a right angle?
GUIDED PRACTICE
Name all the angles in the diagram. Which angle is a right angle?
∠PQR , ∠ PQS, ∠RQS ; ∠ PQS is a right angle .
ANSWER

15. 2. Draw a pair of opposite rays. What type of angle do the rays form?
GUIDED PRACTICE
Draw a pair of opposite rays. What type of angle do the rays form?
ANSWER
Straight Angle

16. Postulate: Protractor Postulate
On a plane, given ray OB and a number between 0° & 180°, there is exactly one ray w/ endpoint O that extends to any side of ray OB such that the degree measure of the angle formed is “r°”. A r° O B

17. Measures of Angles:
The measure of A is denoted by mA. The measure of an angle can be approximated using a protractor, using units called degrees(°). For instance, BAC has a measure of 50°, which can be written as
mBAC = 50°. B C A

18. EXAMPLE 2
Measure and classify angles
Use the diagram to find the measure of the indicated angle. Then classify the angle.
a. ∠KHJ b. ∠GHK c. ∠GHJ d. ∠GHL
SOLUTION
A protractor has an inner and an outer scale. When you measure an angle, check to see which scale to use.

19. Postulate: Angle Addition Postulate
If point D is in the interior of BAC, forming ray AD then:
mBAD + mDAC = mBAC

20. Angle Addition Postulate
EXAMPLE 3
Find angle measures
ALGEBRA Given that m∠LKN =145 , Find m∠LKM & m∠MKN
SOLUTION
Write and solve an equation to find the value of x.
m∠LKN = m∠ LKM + m∠MKN
Angle Addition Postulate
145 = (2x + 10) + (4x – 3) o
Substitute angle measures.
145 = 6x + 7
Combine like terms.
138 = 6x
Subtract 7 from each side.
23 = x
Divide each side by 6.

21. 3. Given that ∠KLM is a straight angle, find m ∠ KLN and m∠ NLM.
GUIDED PRACTICE
Find the indicated angle measures.
Given that ∠KLM is a straight angle, find m ∠ KLN and m∠ NLM.
ANSWERS
125°, 55°

22. GUIDED PRACTICE
4. Given that ∠EFG is a right angle, find m∠EFH and m ∠HFG.
ANSWER
60°, 30°

23. Definition of a Congruent Angles

24. Definition of a Angle Bisector
When a ray lies on the INTERIOR of an angle and divides the angle into 2 congruent angles.
BD is an angle bisector of <ABC. A D B C

25. Ex: If FH bisects EFG & mEFG=120o, find mEFH
m∠GFH + m∠HFE = m∠EFG m∠GFH + m∠GFH = m∠EFG 2 (m∠GFH) = 120° m∠GFH = 60° E H F G

26. Angle Bisector PostulateA D B C

27. Assignment: pp
#1-10 all
12-35 all
45,46
58-64 even