# Solving AAS Triangles

*"AAS" means "Angle, Angle, Side"*

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To solve an AAS triangle

- use the three angles add to 180° to find the other angle
- then The Law of Sines to find each of the other two sides.

### Example 1

In this triangle we know:

- angle A = 35°
- angle C = 62°
- and side c = 7

We can first find **angle B** by using 'angles of a triangle add to 180°':

B = 180° − 35° − 62° = **83°**

To find **side a** we can use The Law of Sines:

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

a/sin(35°) = 7/sin(62°)

Multiply both sides by sin(35°):

a = sin(35°) × 7/sin(62°)

a = **4.55** to 2 decimal places

To find **side b** we can also use The Law of Sines:

b/sin(B) = c/sin(C)

b/sin(83°) = 7/sin(62°)

Multiply both sides by sin(83°):

b = sin(83°) × 7/sin(62°)

b = **7.87** to 2 decimal places

Now we have completely solved the triangle!

We used **b/sin(B) = c/sin(C)** rather than **b/sin(B) = a/sin(A) **for the last calculation ... why?

There's a good reason for that. What if we made a mistake when finding **a**? Then our answer for **b** would also be wrong!

**As a rule, it is always better to use the sides and angles that are given rather than ones we've just worked out.**

### Example 2

This is also an AAS triangle.

First find **angle A** by using "angles of a triangle add to 180°":

**34°**

Now find **side c** by using The Law of Sines:

**8.56**to 2 decimal places

Similarly we can find **side a** by using The Law of Sines and using the given side b = 12.6 (rather than side c that we just worked out):

**7.29**to 2 decimal places

Done!