Rays and Angles Examples
1. In geometry, an angle is defined in terms of two rays that form the angle. You
can think of a ray as a segment that is extended indefinitely in one direction.
Rays have exactly one endpoint, and that point is always named first when
naming the ray.
B
C
E
A
F
D
2. Like segments, rays can also be defined using betweenness of points.
Ray PQ, written PQ , consists of the points on PQ and all points S on PQ such
that Q is between P and S.
3. Any given point on a line determines exactly two rays called opposite rays. This
point is the common endpoint of the opposite rays. In the figure below, PQ and
PR are opposite rays, and P is the common endpoint.
4. Opposite rays can be defined as a figure formed by two collinear rays with a
common endpoint, since the two rays lie on the same line.
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
5. Similarly, an angle can be defined as a figure formed by two rays with a common
endpoint. The two rays are called the sides of the angle. The common endpoint is
called the vertex.
6. The figure formed by opposite rays is often referred to as a straight angle.
Straight angles have a degree measure of 180 degrees.
7. In the figure at the right, the sides of the angle are
YX and YZ , and the vertex is Y. This angle could
be named ∠ Y, ∠ XYZ, ∠ ZYX, or ∠ 1. When
letters are used to name an angle, the letter that
names the vertex is used either as the only letter or
as the middle of three letters.
8. A single letter names an angle only when there is no
chance of confusion. For example, it is not obvious
which angle shown at the right is ∠ A since there
are three different angles that have A as a vertex.
Name the three angles.
∠ BAD, ∠ BAC, and ∠ CAD
9. Example – Whenever two or more angles have a
common vertex, you need to use either three letters
or a number to name each angle. Refer to the figure
at the right to answer each question.
a. What number names ∠ QSP? Î3
b. What is the vertex of ∠ 2? Î Q
c. What are the sides of ∠ 1? Î SQ and SR
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
10. Just as a ruler can be used to measure the length of segment, a protractor can be
used to find the measure of an angle in degrees. To find the measure of an
angle, place the center of the protractor over the vertex of the angle. Then align
the mark labeled 0 on either side of the scale with one side of the angle. This has
been done for ∠ XYZ shown below.
Z
X
Y
Using the inner scale of the protractor, shown in red, you can see that ∠ Y is a 40degree (40o) angle. Thus, we say that the degree measure of ∠ XYZ is 40. This
can also be written as m ∠ XYZ = 40.
11. Protractor Postulate – Given AB and a number r between 0 and 180, there is
exactly one ray with endpoint A, extending on each side of AB , such that the
measure of the angle formed is r.
The protractor postulate guarantees that
there is only one 40o angle on each side of YX .
12. Example – Use a protractor to find the degree measure of each number angle.
m ∠ 1 = 30
m ∠ 2 = 95
m ∠ 3 = 18
m ∠ 4 = 37
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
13. In the figure below, you can see that point R is in the interior of ∠ PQS, m ∠ PQS
= 110 and m ∠ RQS = 30. The sides of ∠ PQR align with the marks labeled 110
and 30 on the inner scale. So m ∠ PQR = 110 – 30 or 80. Since 80 + 30 = 110,
m ∠ PQR + m ∠ RQS = m ∠ PQS. This example and others like it, lead us to the
Angle Addition Postulate.
P
R
Q
14. Angle Addition Postulate – If R is in the interior of ∠ PQS, then m ∠ PQR +
m ∠ RQS = m ∠ PQS. If m ∠ PQR + m ∠ RQS = m ∠ PQS, then R is in the
interior of ∠ PQS.
15. Example – Captain Julie Wright, a pilot for United Airlines, is on approach for a
landing at Chicago’s O’Hare International Airport. Her present compass heading
is 73 degrees. This heading refers to the measurement of the angle formed by the
flight path of the plane and an imaginary path in the direction due north. The
tower has informed Captain Wright to land on runway 9. She knows that by
multiplying the runway number by 10 degrees will give her the compass heading
for a landing on that runway. So the compass heading for her landing must be 90
degrees. How many degrees and in what direction must Captain Wright turn in
order to land on runway 9?
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
S
Let AP represent the path of Captain
Wright’s plane, and let AR represent
the path for a landing on runway 9.
Determine the number of degrees that
the plane must be turned to land on
runway 9 by determining the measure of
∠ PAR.
The compass heading for the path of Captain Wright’s plane, AB , is 73o. Using
the formula given in the problem, we know that the compass heading for a landing
on runway 9 is 9(10) or 90o. AP and AR represent the paths corresponding to
these compass headings. We can use the angle addition postulate to find m ∠ PAR.
m ∠ NAP + m ∠ PAR = m ∠ NAR
73 + m ∠ PAR = 90
m ∠ PAR = 17
Thus, Captain Wright must turn the plane 17o right to land on runway 9.
16. Thought Provoker – Suppose that Captain Wright’s plane was told to land on
runway 6 instead of runway 9. The compass heading for her plane is still 73o.
Determine the number of degrees that the plane must be turned to land on
Runway 6.
m ∠ NAR + m ∠ PAR = m ∠ NAP
60 + m ∠ PAR = 73
m ∠ PAR = 13
Runway 6
Captain Wright must turn the
plane 13o left to land on Runway 6.
Path of plane
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
Name:___________________
Date:____________
Class:___________________
Rays and Angles Activity Sheet
1. What are two other names for QS ?
2. What is the endpoint of SP ?
3. True or false: RX and RT are opposite rays? Why?
4. What are the sides of ∠ 2?
5. Name all of the angles that have RY for a side.
6. Complete: m ∠ XRT = m ∠ 2 + _____.
C
D
B
E
A
P
R
Q
Find the measure of the following angles from above.
7. ∠ PQA
8. ∠ RQE
Rays and Angles
9. ∠ PQC
10. ∠ AQB
11. ∠ BQD
12. ∠ EQC
©2003 www.beaconlearningcenter.com
13. ∠ AQC
14. ∠ AQE
Rev. 09.23.03
4
Use the figure above to answer each question
15. What is the vertex of angle 2?
16. Name a straight angle.
17. Name all the angles that have J as the vertex.
18. Do ∠ 3 and ∠ 4 have a common side? If so, name it.
19. Do ∠ 2 and ∠ J name the same angle? Explain.
In the figure, XP and XT are opposite rays. Given the following conditions, find the
value of x and the measure of the indicated angle.
20. m ∠ SXT = 3x – 4, m ∠ RXS = 2x + 5, m ∠ RXT = 111, find m ∠ RXS.
21. m ∠ PXQ = 2x, m ∠ QXT = 5x – 23, find m ∠ QXT.
22. m ∠ QXR = x + 10, m ∠ QXS = 4x – 1, m ∠ RXS = 91, find m ∠ QXS.
23. m ∠ QXR = 3x + 5, m ∠ QXP = 2x – 3, m ∠ RXP = x + 50, find m ∠ RXT.
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
Name:___________________
Date:____________
Class:___________________
Rays and Angles Activity Sheet Key
1. What are two other names for QS ? Î QR , QT
2. What is the endpoint of SP ? Î S
3. True or false: RX and RT are opposite rays? Why? Î No, the rays do not lie on
the same line.
4. What are the sides of ∠ 2? Î RX , RY
5. Name all of the angles that have RY for a side. Î ∠ PRY, ∠ XRY, ∠ TRY
6. Complete: m ∠ XRT = m ∠ 2 + _____. Î m ∠ 3
D
C
B
E
A
R
P
Q
Find the measure of the following angles from above. (Answers are
approximations.)
7. ∠ PQAÎ 10o
8. ∠ RQEÎ 25o
9. ∠ PQCÎ 105o
Rays and Angles
10. ∠ AQBÎ 30o
11. ∠ BQDÎ 80o
12. ∠ EQCÎ 50o
©2003 www.beaconlearningcenter.com
13. ∠ AQCÎ 95o
14. ∠ AQEÎ 145o
Rev. 09.23.03
4
Use the figure above to answer each question.
15. What is the vertex of angle 2? Î J
16. Name a straight angle. Î ∠ HUK
17. Name all the angles that have J as the vertex. Î ∠ HJK, ∠ HJU, ∠ UJK
18. Do ∠ 3 and ∠ 4 have a common side? If so, name it.Î No
19. Do ∠ 2 and ∠ J name the same angle? Explain.Î No, since ∠ J could refer to
∠ 1, ∠ 2, or ∠ HJK.
In the figure, XP and XT are opposite rays. Given the following conditions, find the
value of x and the measure of the indicated angle.
20. m ∠ SXT = 3x – 4, m ∠ RXS = 2x + 5, m ∠ RXT = 111, find m ∠ RXS.
∠ SXT + ∠ RXS = ∠ RXT Î (3x – 4) + (2x + 5) = 111
5x + 1 = 111 Î 5x = 110 Î x = 22
∠ RXS = 2(22) + 5 Î 49o
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
21. m ∠ PXQ = 2x, m ∠ QXT = 5x – 23, find m ∠ QXT.
m ∠ PXQ + ∠ QXT = 180Î(2x) + (5x – 23) = 180
7x – 23 = 180 Î 7x = 203 Î x = 29
m ∠ QXT = 5(29) – 23 = 122o
22. m ∠ QXR = x + 10, m ∠ QXS = 4x – 1, m ∠ RXS = 91, find m ∠ QXS.
m ∠ QXR + ∠ RXS = ∠ QXS Î(x + 10) + (91) = (4x – 1)
x + 101 = 4x – 1 Î 102 = 3x Î x = 34
∠ QXS = 4(34) – 1 = 135o
23. m ∠ QXR = 3x + 5, m ∠ QXP = 2x – 3, m ∠ RXP = x + 50, find m ∠ RXT.
m ∠ QXR + m ∠ QXP = m ∠ RXP Î(3x + 5) + (2x – 3) = (x + 50)
5x + 2 = x + 50 Î 4x = 48 Î x = 12
m ∠ RXT + m ∠ RXP = 180Îm ∠ RXT + (12 + 50) = 180
m ∠ RXT = 118o
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
Student Name: __________________
Date: ______________
Rays and Angles Checklist
(This is a suggested checklist if you are using this as a number grade; on the other hand,
you could devise your own rubric.)
1.
On question 1, did the student give two other names for QS ?
a.
b.
2.
On question 2, did the student give the correct endpoint?
a.
3.
All eleven (55 points)
Ten of the eleven (50 points)
Nine of the eleven (45 points)
Eight of the eleven (40 points)
Seven of the eleven (35 points)
Six of the eleven (30 points)
Five of the eleven (25 points)
Four of the eleven (20 points)
Three of the eleven (15 points)
Two of the eleven (10 points)
One of the eleven (5 points)
On question 17, did the student name all the angles?
a.
b.
c.
8.
All three (15 points)
Two of the three (10 points)
One of the three (5 points)
On questions 6 thru 16, did the student answer questions correctly?
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
7.
Both (10 points)
One of two (5 points)
On question 5, did the student name all the angles?
a.
b.
c.
6.
Yes (10 points)
Student answered correctly but did not describe “why” (5 points)
On question 4, did the student give correct sides?
a.
b.
5.
Yes (5 points)
On question 3, did the student answer (all) parts of the question correctly?
a.
b.
4.
Yes (10 points)
Student gave one correct name (5 points)
All three (15 points)
Two of the three (10 points)
One of the three (5 points)
On question 18, did the student answer question correctly?
a.
Yes (5 points)
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03
9.
On question 19, did the student answer (all) parts of the question correctly?
a.
b.
Yes (10 points)
Yes, but did not explain
10. On question 20, did the student find the value of x and the measure of the missing angle?
a.
b.
Yes (10 points)
Found the value of x but not the missing angle (5 points)
11. On question 21, did the student find the value of x and the measure of the missing angle?
a.
b.
Yes (10 points)
Found the value of x but not the missing angle (5 points)
12. On question 22, did the student find the value of x and the measure of the missing angle?
a.
b.
Yes (10 points)
Found the value of x but not the missing angle (5 points)
13. On question 23, did the student find the value of x and the measure of the missing angle?
a.
b.
Yes (10 points)
Found the value of x but not the missing angle (5 points)
Total Number of Points _________
A 162 points and above
Any score below C
needs
remediation!
B 144 points and above
C 126 points and above
D 108 points and above
F
107 points and below
Rays and Angles
©2003 www.beaconlearningcenter.com
Rev. 09.23.03