Supplementary and Complementary Angles - Sonlight
LESSON 6
Supplementary and Complementary
Angles
LESSON 6
Supplementary and Complementary Angles
Greek Letters
¦Á
Figure 1
¦Á = alpha
¦Â
¦Â = beta
¦Ã
¦Ä
¦Ã = gamma
¦Ä = delta
Adjacent Angles - Angles that share a common side and have the same origin
are called adjacent angles. They are side by side. In figure 1, ¦Á is adjacent to both ¦Â
and ¦Ä. It is not adjacent to ¦Ã. In figure 1, there are four pairs of adjacent angles: ¦Á
and ¦Â, ¦Â and ¦Ã, ¦Ã and ¦Ä, ¦Ä and ¦Á.
In figure 2, we added points so we can name the rays that form the angles. The
common side shared by adjacent angles ¦Á and ¦Â is VQ.
Q
¦Á
Figure 2
T
¦Â
V
¦Ä
R
? ?
Given: RT ¡É QS = V
¦Ã
S
Vertical Angles - Notice that ¡Ï¦Ã is opposite ¡Ï¦Á. Angles that share a common
origin and are opposite each other are called vertical angles. They have the same
measure and are congruent. ¡Ï¦Â and ¡Ï¦Ä are also vertical angles.
GEOMETRY
SUPPLEMENTARY AND COMPLEMENTARY ANGLES - LESSON 6
37
Figure 2 (from previous page)
Q
¦Á
R
¦Â
V
? ?
Given: RT ¡É QS = V
¦Ã
¦Ä
T
S
If m¡Ï¦Â is 115¡ã, then m ¡Ï¦Ä is also 115¡ã. If this is true, then do we have enough
information to find m ¡Ï¦Á? We know from the information given in figure 2 that
?
?
RT and QS are lines. Therefore, ¡ÏRVT is a straight angle and has a measure of
180¡ã. If ¡ÏRVQ (¡Ï¦Â) is 115¡ã, then ¡ÏQVT (¡Ï¦Á) must be 180¡ã - 115¡ã, or 65¡ã. Since
¡ÏRVS (¡Ï¦Ã) is a vertical angle to ¡ÏQVT, then it is also 65¡ã.
Supplementary Angles - Two angles such as ¡Ï¦Á and ¡Ï¦Â in figure 2, whose
measures add up to 180¡ã, or that make a straight angle (straight line), are said to be
supplementary. In figure 2, the angles were adjacent to each other, but they don't
have to be adjacent to be classified as supplementary angles.
Figure 3
D
6
G
?
K
5
?
J
?
1
H
4
?
2
3
?
E
All drawings are in
the same plane unless
otherwise noted.
?
Given: DF, GE, and KJ all
?F
intersect at H.
DF
¡Í GE
Complementary Angles - We can observe many relationships in figure 3.
¡Ï1 is adjacent to both ¡Ï6 and ¡Ï2. Angle 3 and ¡Ï6 are vertical angles, as are ¡Ï1
? ?
and ¡Ï4. Angle 6 and ¡Ï3 are also right angles since DF ¡Í GE. The new concept
here is the relationship between ¡ÏDHE and ¡ÏGHF. Both of these are right angles
because the lines are perpendicular; therefore their measures are each 90¡ã. Then
m¡Ï1 + m¡Ï2 = 90¡ã, and m¡Ï4 + m¡Ï5 = 90¡ã. Two angles whose measures add up
to 90¡ã are called complementary angles. Notice that from what we know about
vertical angles, ¡Ï1 and ¡Ï5 are also complementary. Let's use some real measures
to verify our conclusions.
38
LESSON 6 - SUPPLEMENTARY AND COMPLEMENTARY ANGLES
GEOMETRY
Figure 4
(a simplified figure 3)
1
2
5
4
In figure 4, let's assume that m¡Ï1 = 47¡ã. Then m¡Ï2 must be 43¡ã, since m¡Ï1
and m¡Ï2 add up to 90¡ã. If m¡Ï1 = 47¡ã, then m¡Ï4 must also be 47¡ã, since ¡Ï1 and
¡Ï4 are vertical angles. Also, m¡Ï5 must be 43¡ã. So ¡Ï1 and ¡Ï5 are complementary,
as are ¡Ï2 and ¡Ï4. Remember that supplementary and complementary angles do
not have to be adjacent to qualify.
It helps me to not get supplementary and complementary angles mixed up if I
think of the s in straight and the s in supplementary. The c in complementary may
be like the c in corner.
GEOMETRY
SUPPLEMENTARY AND COMPLEMENTARY ANGLES - LESSON 6
39
6A
lesson practice
Use the drawing to fill in the blanks.
1. ¡ÏAHC is adjacent to ¡Ï______ and ¡Ï______.
2. ¡ÏBHD is adjacent to ¡Ï______ and ¡Ï______.
3. ¡ÏFHB and ¡Ï______
are vertical angles.
J
4. ¡ÏFHC and ¡Ï______
are vertical angles.
L
C
F
K
B
A
H
5. ¡ÏLFJ and ¡Ï______
are supplementary angles.
D
G
6. ¡ÏFHC and ¡Ï______
are complementary angles.
? ? ?
?
Given: AB, CD, LG, and JK are
straight lines. m¡ÏFHB = 90?.
7. ¡ÏJFH and ¡Ï______
are supplementary angles.
8. ¡ÏBHD and ¡Ï______
are complementary angles.
Q
T
The drawing is a sketch and not
necessarily to scale. Don¡¯t make
P
M
N
any assumptions about the lines
S
and angles
other than what is
actually
given.
X
R
Z
9. If m¡ÏCHA = 40?, then m¡ÏBHD = ______.
GEOMETRY Lesson Practice 6A
Y
63
LESSON PRACTICe 6A
Use the drawing from the previous page to fill in the blanks.
10. If m¡ÏJFL = 65?, then m¡ÏKFH = ______.
11. If m¡ÏFHB = 90?, then m¡ÏFHA = ______.
12. If m¡ÏCHA = 40?, then m¡ÏFHC = ______.
13. If m¡ÏLFJ = 65?, then m¡ÏLFK = ______.
14. If m¡ÏFHB = 90?, then m¡ÏAHG = ______.
Use the letters to match each term to the best answer.
15. ¦Â ___
a. share a common ray
16. adjacent angles ___
b. alpha
17. supplementary angles ___
c. always have the same measure
18.
64
¦Á ___
d. add up to 90?
19. complementary angles ___
e. add up to 180?
20. vertical angles ___
f. beta
GEOMETRY
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