1. 1.4 What you should learn Why you should learn it
Angles and Their Measures
What you should learn
GOAL 1
Use Angle Postulates
GOAL 2
Classify angles as acute, right, obtuse, or straight.
Why you should learn it
To solve real-life problems about angles, such as the field of vision of a horse wearing blinkers.

2. 1.4 Angles and Their Measures 1 GOAL USING ANGLE POSTULATES
An _____ consists of two different rays that have the same initial point.
angle
The rays are the _____ of the angle and the initial point is the ______.
sides
vertex
The three names for this angle are A C B
The sides are
The vertex is point A.

3. Name the three angles in the diagram.
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 X as their vertex.

4. Extra Example 1 Name the angles in the figure. M N O P
Click for the answers.
should not be used to name any angle in the figure. Why not? 1. 2. 3.
All three angles have N as the vertex, so could mean any of the angles.

5. Measuring Angles The expression is read as “the ________ of angle A.”
measure
IMPORTANT!!! Note the difference in notation between an angle and its measure. Always use the correct notation!!!
The tool used to measure angles is called a _________.
protractor
The units used to measure angles are called _______, and the symbol for them is a _.
degrees

6. Measure and classify angles
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.

7. Measure and classify angles
EXAMPLE 2
Measure and classify angles
a. HJ is lined up with the 0 on the inner scale of the protractor. HK passes through 55 on the inner scale. So, m KHJ = 55 . It is an acute angle. o
b. HG is lined up with the 0 on the outer scale and HK passes through 125 on the outer scale. So, m GHK = It is an obtuse angle. o
c. m GHJ = 180.
It is a straight angle. o
d. m GHL= 90.
It is a right angle. o

8. Measuring Angles Q R S Let’s measure some angles. Y M T
Since we say that the angles are _________.
congruent
Remember: Angles are congruent, measures are equal.
Incorrect:
Correct:
and

9. TAKE NOTE! MEASURES ARE EQUAL mBAC = mDEF ANGLES ARE CONGRUENT
BAC  DEF
“is congruent to”
“is equal to”
Note that there is an m in front when you say equal to; whereas the congruency symbol  ; you would say congruent to. (no m’s in front of the angle symbols).

10. PROTRACTOR POSTULATE
For any point A on one side of , can be matched one to one with the real numbers from 0 to ___.
180
The absolute value of the difference between the real numbers for is the ________ of
measure
Check your understanding of the Protractor Postulate by finding Use either scale on the protractor to find it, but use the same one for both rays. B O A
Click to see two solutions. or

11. To understand the next postulate, you must understand some vocabulary:
A point that is between points that lie on each side of an angle is in the _______ of the angle.
A point that is not on an angle or in its interior is in the ________ of the angle.
interior
exterior
exterior B
interior A Z
In the above diagram, A is in the interior of the angle and B is in the exterior of the angle.

12. ANGLE ADDITION POSTULATE
If P is in the interior of
then R T P S
Does it make sense? Study the figure
to be sure you understand before going on!
EXAMPLE 2

13. What is the measure of the angle that is shaded green?
Extra Example 2
The backyard of a house is illuminated by a light fixture that has two bulbs. Each bulb illuminates an angle of 120°. If the angle illuminated only by the right bulb is 35°, what is the angle illuminated by both bulbs?
Click for a picture.
What is the measure of the angle that is shaded green?
120°
Click for the solution.
120°
120°
- 35° =
85°
85°
Do you see the application of the Angle Addition Postulate?
35°

14. Checkpoint D Name the angles in the figure. Click for the solution. C
In the figure above, and
Find the measure of Click for the solution.

15. 1.4 Angles and Their Measures 2 GOAL CLASSIFYING ANGLES
We can classify all angles by their measure as follows:
Angles with measures between 0 and 90 degrees: _____
90 degree angles: _____
Angles with measures between 90 and 180 degrees: ______
180 degree angles: _______
acute
right
obtuse
straight

16. Can you match each angle with its description?
Click to check your answer. A
Important!!! Anytime this box appears at the vertex of an angle, it means that the angle is a right angle. You MUST use it when you want to show a right angle.
acute:
right:
obtuse:
straight: A A A
EXAMPLE 3

17. Classifying Angles in a Coordinate Plane
Plot the points L(-4,2), M(-1,-1), N(2,2), Q(4,-1), and P(2,-4). Then measure and classify the following angles as acute, right, obtuse, or straight.
LMN
LMP
NMQ
LMQ

18. Solution:
Begin by plotting the points. Then use a protractor to measure each angle.

19. Solution:
Begin by plotting the points. Then use a protractor to measure each angle.

20. Extra Example 3
Plot the points A(-3, -1), B(-1, 1), C(2, 4), D(2, 1), and E(2, -2).
Then measure and classify the following angles as acute, right, obtuse, or straight. Click for each answer. a. b. c. d.
45°, acute
90°, right
180°, straight
135°, obtuse
Reminder: Be sure you understand the material before going on to the next slide. If you need to review, do so NOW!

21. 1. Name all the angles in the diagram. Which angle is a right angle?
GUIDED PRACTICE
for Examples 1and 2
Name all the angles in the diagram. Which angle is a right angle?
PQR , PQS, RQS ; PQS is a right angle .
ANSWER

22. 2. Draw a pair of opposite rays. What type of angle do the rays form?
GUIDED PRACTICE
for Examples 1and 2
Draw a pair of opposite rays. What type of angle do the rays form?
ANSWER
Straight Angle

23. Angle Addition Postulate Substitute angle measures.
EXAMPLE 3
Find angle measures o
ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
SOLUTION
STEP 1
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.

24. Evaluate the given expressions when x = 23.
EXAMPLE 3
Find angle measures
STEP 2
Evaluate the given expressions when x = 23.
m LKM = (2x + 10)° = ( )° = 56°
m MKN = (4x – 3)° = (4 23 – 3)° = 89°
So, m LKM = 56° and m MKN = 89°.
ANSWER

25. Find the indicated angle measures.
GUIDED PRACTICE
for Example 3
Find the indicated angle measures.
Given that KLM is a straight angle, find m KLN and m NLM.
ANSWER
125°, 55°

26. 4. Given that EFG is a right angle, find m EFH and m HFG.
GUIDED PRACTICE
for Example 3
4. Given that EFG is a right angle, find m EFH and m HFG.
ANSWER
60°, 30°

27. Identify congruent angles
EXAMPLE 4
Identify congruent angles
The diagram shows some of the angles formed by the ropes in a trapeze apparatus. Identify the congruent angles. If m DEG = 157° ,what is m GKL?
Trapeze
SOLUTION
There are two pairs of congruent angles:
DEF JKL and DEG GKL. ~
Because DEG GKL, m DEG = m GKL.
So, m GKL = 157°. ~

28. 5. Identify all pairs of congruent angles in the diagram.
GUIDED PRACTICE
for Example 4
Use the diagram shown.
5. Identify all pairs of congruent angles in the diagram.
T and S, P and R.
ANSWER

29. GUIDED PRACTICE for Example 4 Use the diagram shown.
6. In the diagram, m PQR = 130 , m QRS = 84, and m TSR = Find the other angle measures in the diagram. o
m PTS = 121,
m QPT = 84°
ANSWER

30. An ANGLE BISECTOR is a ray that divides an angle into two angles that are congruent.
Draw and label an acute angle,
Fold the paper so that MN is on top of LM.
Draw a point P on the folded side. Connect it with M.
Measure the angles formed.

31. Double an angle measure
EXAMPLE 5
Double an angle measure
In the diagram at the right, YW bisects XYZ, and m XYW = 18 . Find m XYZ. o
SOLUTION
By the Angle Addition Postulate, m XYZ = m XYW + m WYZ. Because YW bisects XYZ you know that XYW WYZ. ~
So, m XYW = m WYZ, and you can write
m XYZ = m XYW + m WYZ = 18°+ 18° = 36°.

32. GUIDED PRACTICE
for Example 5
7. Angle MNP is a straight angle, and NQ bisects MNP. Draw MNP And NQ . Use arcs to mark the congruent angles in your diagram, and give the angle measures of these congruent angles.
90°
ANSWER

33. Classify each angle as acute, obtuse , right or straight.
Daily Homework Quiz
Classify each angle as acute, obtuse , right or straight. o
1. m A = 90
ANSWER
right
2. m B = 62 o
ANSWER
acute
3. m C = 119 o
ANSWER
obtuse

34. 5. If XY bisects ZXW and m XZY = 36 , find m ZXW.
Daily Homework Quiz
If m DEG = 84 , find m FEG. o
ANSWER 14 o
5. If XY bisects ZXW and m XZY = 36 , find m ZXW. o
ANSWER 72 o

35. The common vertex is __ and the common side is ___.
Two angles that share a common vertex and side, but have no common interior points, are called ________ angles.
adjacent A B C D
Name the two adjacent angles in the diagram. Click to check.
The common vertex is __ and the common side is ___. D
EXAMPLE 4

36. Classifying Angles in a Coordinate Plane
Plot the points L(-4,2), M(-1,-1), N(2,2), Q(4,-1), and P(2,-4). Then measure and classify the following angles as acute, right, obtuse, or straight.
LMN
LMP
NMQ
LMQ

37. Solution:
Begin by plotting the points. Then use a protractor to measure each angle.

38. Solution:
Begin by plotting the points. Then use a protractor to measure each angle.
Two angles are adjacent angles if they share a common vertex and side, but have no common interior points.

39. Extra Example 4 Use a protractor to draw two adjacent angles and
so that is acute and is straight.
Click to see a sample answer. L N M O
Classify as acute, right, obtuse, or straight:
obtuse

40. Does your solution meet the requirements above?
Checkpoint
Draw 5 points A, B, C, D, and E so that all four statements are true:
are adjacent.
is obtuse.
D is in the exterior of
is a right angle.
Click to see a sample answer. B C E A D
Does your solution meet the requirements above?

41. QUESTIONS?