Ambiguous Case
In ∆ABC, m<A = 30°, a = 7, c = 12, find m<C.
SinA SinC

a
c
Since Sin was positive and Sin is positive in both
Quadrants I and II, so we can actually have 2 triangles
Sin 30 SinC

7
12
QI would be 59°, 30° and 91°
m<C = 59°
QII we would have to subtract 59 from 180 bc 59 = RA
so <C = 180 – 59 = 121°
therefore QII angles can be 121°, 30° and 29°
The “Ambiguous Case” is a term used when we want to determine the number of
possible triangles that can be constructed when we are given 2 sides and angle opposite
of them (SSA)
Helpful relationships when given SSA and need to determine the # of triangles
- If given <A is obtuse and a < c, 0 triangles can be formed
- If given <A is obtuse and a > c, 1 triangle can be formed
- If given <A is acute and c > a > c(sinA), 2 triangles can be formed
- If given <A is acute and a = c(sinA), 1 triangle can be formed
- If given <A is acute and a > c > c(sinA), 1 triangle can be formed
- If given <A is acute and a < c(sinA), 0 triangles can be formed
Example Questions:
Decide the number of triangles that can be formed in ∆ABC
1) a = 10, b = 12, m<B = 20°
SinA SinB

a
b
SinA Sin 20

10
12
QI : <A = 16.6
QII: <A = 180 – 16.6 = 163.4
2) a = 12, c = 31, m<A = 17°
SinA SinC

a
c
Sin17 SinC

12
31
QI : <C = 49
QII: <A = 180 – 49 = 131
Then subtract the 2 angles that you know from 180°. If the answer is between 0 and 180
a triangle can be formed from those 3 angles
QI : 180 – (20 + 16.6) = 143.4 ☺
QI : 180 – (17 + 49) = 114
☺
☺
QII : 180 – (20 + 163.4) = 183.4 REJECT
Only ONE triangle can be formed
QII : 180 – (17 + 131) = 32
TWO triangles can be formed
3) a = 4, c =12, m<A = 30°
SinA SinC

a
c
Sin 30 SinC

4
12
6
4
Reject because sin cannot
Be greater than one.
Sin C =
NO triangles can be formed
1)
2)
3)
4)
5)
6)
7)
Additional practice
Answers
a = 4, b = 6, m<A = 30°
a = 4, b = 6, m<A = 150°
a = 4, b = 6, m<B = 150°
a = 15, b = 12, m<B = 45°
a = 15, b = 12, m<A = 135
a = 5, b = 8, m<A = 40°
a = 9, b = 12, m<A = 35°
- 2 ∆’s
- 0 ∆’s
-1∆
- 2 ∆’s
-1∆
- 0 ∆’s
- 2 ∆’s
8) How many distinct triangles can be constructed if the measure of two sides are to be
35 and 70 and the measure of the angle opposite the smaller of these sides is to be 30?
Ans: 0 triangles