Solving AAS Triangles

"AAS" means "Angle, Angle, Side"

AAS Triangle

This means we are given two angles and one side (which is not between the angles).

 

To solve an AAS triangle

 

Example 1

In this triangle we know:

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

 

It's easy to find angle B by using 'angles of a triangle add to 180°':

B = 180° - 35° - 62° = 83°

 

We can also find side a by using The Law of Sines:

a/sin A = c/sin C

a/sin(35°) = 7/sin(62°)

a = (7 × sin(35°))/sin(62°) = 4.55 (to 2 decimal places).

 

Also we can find b by using The Law of Sines:

b/sin B = c/sin C

b/sin(83°) = 7/sin(62°)

b = (7 × sin(83°))/sin(62°) = 7.87 to 2 decimal places.

 

Now we have completely solved the triangle!

Did you notice that we used b/sin B = c/sin C rather than b/sin B = a/sin A for the last calculation?

There's a good reason for that. What if we had made a mistake in 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°':

A = 180° - 41° - 105° = 34°

 

Now find side c by using The Law of Sines:

c/sin C = b/sin B
c/sin(41°) = 12.6/sin(105°)
c = (12.6 × sin(41°))/sin(105°) = 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 c that we just worked out:

a/sin A = b/sin B
a/sin(34°) = 12.6/sin(105°)
a = (12.6 × sin(34°))/sin(105°) = 7.29 to 2 decimal places.

Done!