Supplementary Angles
Supplementary angles are two angles whose measures add up to
180
°
.
The two angles of a linear pair , like
∠
1
and
∠
2
in the figure below, are always supplementary.
But, two angles need not be adjacent to be supplementary. In the next figure,
∠
3
and
∠
4
are supplementary, because their measures add to
180
°
.
Example 1:
Two angles are supplementary. If the measure of the angle is twice the measure of the other, find the measure of each angle.
Let the measure of one of the supplementary angles be
a
.
Measure of the other angle is
2
times
a
.
So, measure of the other angle is
2
a
.
If the sum of the measures of two angles is
180
°
, then the angles are supplementary.
So,
a
+
2
a
=
180
°
Simplify.
3
a
=
180
°
To isolate
a
, divide both sides of the equation by
3
.
3
a
3
=
180
°
3
a = 60 °
The measure of the second angle is,
2 a = 2 × 60 ° = 120 °
So, the measures of the two supplementary angles are
60
°
and
120
°
.
Example 2:
Find
m
∠
P
and
m
∠
Q
if
∠
P
and
∠
Q
are supplementary,
m
∠
P
=
2
x
+
15
, and
m
∠
Q
=
5
x
−
38
.
The sum of the measures of two supplementary angles is
180
°
.
So,
m
∠
P
+
m
∠
Q
=
180
°
Substitute
2
x
+
15
for
m
∠
P
and
5
x
−
38
for
m
∠
Q
.
2
x
+
15
+
5
x
−
38
=
180
°
Combine the like terms. We get:
7
x
−
23
=
180
°
Add
23
to both the sides. We get:
7
x
=
203
°
Divide both the sides by
7
.
7
x
7
=
203
°
7
Simplify.
x
=
29
°
To find
m
∠
P
, substitute
29
for
x
in
2
x
+
15
.
2
(
29
)
+
15
=
58
+
15
Simplify.
58
+
15
=
73
So,
m
∠
P
=
73
°
.
To find
m
∠
Q
, substitute
29
for
x
in
5
x
−
38
.
5
(
29
)
−
38
=
145
−
38
Simplify.
145
−
38
=
107
So,
m
∠
Q
=
107
°
.