# Solving SSA Triangles

*"SSA" means "Side, Side, Angle"*

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To solve an SSA triangle

- use The Law of Sines first 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 1

In this triangle we know

- angle B = 31°
- b = 8
- and c = 13

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

^{−1}(0.8369...)

**56.8°**to one decimal place

Next, we can use the three angles add to 180° to find angle A:

**92.2°**to one decimal place

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

Notice that we didn't use A = 92.2°, that angle is rounded to 1 decimal place. It's much better to use the unrounded number 92.181...° which should still be on our calculator from the last calculation.

**15.52**to 2 decimal places

So, we have completely solved the triangle ...

... or have we?

* Back when we calculated:

C = sin^{−1}(0.8369...)

C = 56.818...°

We didn't include that **sin ^{−1}(0.8369...)** might have

**two answers**(see Law of Sines):

The other answer for C is **180° − 56.818...°**

Here you can see why we have two possible answers:

By swinging side "8" left and right we can

join up with side "a" in two possible locations.

So let's go back and continue our example:

The other possible angle is:

**123.2°**to one decimal place

With a new value for C we will have new values for angle **A** and side **a**

Use "the three angles add to 180°" to find angle A:

**25.8°**to one decimal place

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

**6.76**to 2 decimal places

So the two sets of answers are:

C = 56.8°, A = 92.2°, a = 15.52

C = 123.2°, A = 25.8°, a = 6.76

### Example 2

This is also an SSA triangle.

In this triangle we know angle M = 125°, m = 12.4 and l = 7.6

We will use The Law of Sines to find angle L first:

**30.1°**to one decimal place

Next, we will use "the three angles add to 180°" to find angle N:

**24.9°**to one decimal place

Now we will use The Law of Sines again to find n:

**6.36**to 2 decimal places

**Note** there is only one answer in this case. The "12.4" line only joins up one place.

The other *possible* answer for L is 149.9°. But that is **impossible** because we already have M = 125° and a triangle can't have two angles greater than 90°.

## Conclusion:

When solving a *"Side, Side, Angle"* triangle we start with the Law of Sines, but need to check if there is another possible answer!