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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