![](http://s3.amazonaws.com/prod-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10849.jpg)
Figure 1
Two cases for SSA.
![](http://s3.amazonaws.com/prod-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10850.jpg)
Figure 2
Ambiguous cases for SSA.
![](http://s3.amazonaws.com/prod-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10851.jpg)
Figure 3
Two cases for SSA.
In Figure 1(a), if b < h, then b cannot reach the other side of the triangle, and no solution is possible. This occurs when b < a sin β.
In Figure 1 (b), if b = h = a sin β, then exactly one right triangle is formed.
In Figure 2 (a), if h < b < a—that is, a sin β < b, < a—then two different solutions exist.
In Figure 2 (b), if b = a, then only one solution exists, and if b = a, then the solution is an isosceles triangle.
If β is an obtuse or right angle, the following two possibilities exist.
In Figure 3 (a), if b > a, then one solution is possible.
In Figure 3 (b), if b ≤ a, then no solutions are possible.
Example 1: (SSS) Find the difference between the largest and smallest angles of a triangle if the lengths of the sides are 10, 19, and 23, as shown in Figure 4 .
![](http://s3.amazonaws.com/prod-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10852.jpg)
Figure 4
Drawing for Example 1.
First, use the Law of Cosines to find the size of the largest angle (β) which is opposite the longest side (23).
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10811.jpg)
Next, use the Law of Sines to find the size of the smallest angle (α), which is opposite the shortest side (10).
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10812.jpg)
Thus, the difference between the largest and smallest angle is
![](https://s3.amazonaws.com/dev-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10813.jpg)
Example 2: (SAS) The legs of an isosceles triangle have a length of 28 and form a 17° angle (Figure 5). What is the length of the third side of the triangle?
![](http://s3.amazonaws.com/prod-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10853.jpg)
Figure 5
Drawing for Example 2.
This is a direct application of the Law of Cosines.
Example 3: (ASA) Find the value of d in Figure 6 .
![](http://s3.amazonaws.com/prod-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10854.jpg)
Figure 6
Drawing for Example 3.
First, calculate the sizes of angles α and β. Then find the value of a using the Law of Sines. Finally, use the definition of the sine to find the value of d.
Finally,
Example 4: (AAS) Find the value of x in Figure 7 .
![](http://s3.amazonaws.com/prod-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10855.jpg)
Figure 7
Drawing for Example 4.
First, calculate the size of angle α. Then use the Law of Sines to calculate the value of x.
Example 5: (SSA) One side of a triangle, of length 20, forms a 42° angle with a second side of the triangle (8). The length of the third side of the triangle is 14. Find the length of the second side.
![](http://s3.amazonaws.com/prod-hmhco-vmg-craftcms-public/_cliffsnotes/assets/10856.jpg)
Figure 8
Drawing for Example 5.
The length of the altitude (h) is calculated first so that the number of solutions (0, 1, or 2) can be determined.
Because 13.38 < 14 < 20, there are two distinct solutions.
Solution 1: Use of the Law of Sines to calculate α.
Use the fact that there are 180° in a triangle to calculate β
Use the Law of Sines to find the value of b.
Solution 2: Use α to find α′, and α′ to find β′
Next, use the Law of Sines to find b′.