## Solving SSA Triangles

SSA (Side, Side, Angle) is when we know 2 sides and 1 angle that is not the angle between the sides.

## How to solve an SSA triangle ?

First, use Law of Sines to calculate one of the other two angles.

Then, use the three angles add to 180° to find the other angle.

Finally use The Law of Sines again to find the unknown side.

### Example

Find the unknown side and angles

### Answer

From this triangle we get

angle *A* = 31°

*a* = 8 and

c = 13

In this case, we can use The Law of Sines first to find angle C:

sin(*C*)/*c* = sin(*A*)/*a*

sin(*C*)/13 = sin(31°)/8

sin(*C*) = (13×sin(31°))/8

sin(*C*) = 0.8369…

*C* = sin^{−1}(0.8369…)

*C* = 56.818…°

*C* = 56.8° to one decimal place

Then, use the 3 angles add to 180° to find angle *B*:

*B* = 180° − 31° − 56.818…°

*B* = 92.181…° = 92.2° to one decimal place

Next, use The Law of Sines again to find *b*:

*b*/sin(*B*) = *a*/sin(*A*)

*b*/sin(92.181…°) = 8/sin(31°)

We didn’t use *B* = 92.2°, (the angle is rounded to 1 decimal place).

It’s better to use the unrounded number 92.181…° which still be on our calculator from the last calculation.

*b* = (sin(92.181…°) × 8)/sin(31°)

*b* = 15.52 to 2 decimal places

### Other possible answer?

Back to:

C = sin^{−1}(0.8369…)

C = 56.818…°

sin^{−1}(0.8369…) have two answers

The other answer for C is 180° − 56.818…°

The other possible angle is:

C = 180° − 56.818…°

C = 123.2° to one decimal place

With a different value for C we will have different values for angle B and side *b*

Use the 3 angles add to 180° to find angle *B*:

*B* = 180° − 31° − 123.181…°

*B* = 25.818…°

*B* = 25.8° to one decimal place

Now we can use The Law of Sines again to find *b*:

*b*/sin(*B*) = *a*/sin(*A*)

*b*/sin(25.818…°) = 8/sin(31°)

*b* = (sin(25.818…°)×8)/sin(31°)

*b* = 6.76 to 2 decimal places

So the 2 sets of answers are:

C = 56.8°, B = 92.2°, *b* = 15.52

C = 123.2°, B = 25.8°, *b* = 6.76