What Is An SSA Triangle?

SSA stands for side – side – angle, and it means a triangle with known lengths of two sides and one known angle that is not between the two sides. When we have this much information, we can figure out the remaining side and the other two angles.

Describing An SSA Triangle

Picture a triangle with sides a, b, and c. The angle opposite side a is angle A, the angle opposite side b is angle B, and the angle opposite side c is angle C. In this triangle, we know that side a is 8 inches long, and side b is 6 inches long. We also know that angle A is 50 degrees.

Step One: Find Another Angle

We can use the Law of Sines to find either one of the unknown angles in the triangle. The Law of Sines states these ratios are equal:

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

When we use this law to find an unknown angle, it is easier to flip the equations, like this:

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

Now, we can plug in the information that we have.

sin(50)/8 = sin(B)/6

Since we have no information about side c or Angle C at this point, we can ignore them. Using algebra, we can rewrite the equation as

sin(B) = (sin(50)/8) x 6

Using a calculator, we can figure out that

sin(B) = .77/8 x 6 = .58

So now we know that the sine of angle B is equal to .58, but we have to take one more step to find out what angle B actually is. The next step is to calculate the inverse sine (sin-1). The inverse sine of .58 is 35.5, so angle B is 35.5 degrees.

Step Two: Find The Third Angle.

Now that we know two angles in the triangle, it’s very easy to find the third. We just need to remember the rule that all three angles of any triangle add up to 180. We know that angle A = 50 and angle B = 35.5; therefore, angle C = 94.5

Step Three: Find The Other Side.

Now we can go back to the Law of Sines to find the length of side c. Since we know two sides and their opposite angles, we can use either pair. For this example, we’ll use a/sin(A).

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

Plugging in the information we have, we get the equation

8/sin(50) = c/sin(94.5)

or

c = 8/sin(50) x sin(94.5) = 10.4 x 1 = 10.4 inches

Note that there can sometimes be two possible answers for an SSA triangle. If the known angle is less than 90 degrees, it could be possible to flip the triangle, using the same side values but a supplementary angle, since the sine of any angle and the sine of its supplement are the same. This ambiguity is only possible when the known angle is less than 90, the adjacent side is longer than the opposite side, and the length of the opposite side is greater than the height.

Practice

Given the following information, find angles B and C and side c.

side a = 15

side b = 10

side c = ?

angle A = 120

angle B = ?

angle C = ?

Solution:

First find angle B using the Law of Sines.

sin(120)/15 = sin(B)/10

sin(B) = sin(120)/15 x 10 = .58

sine-1(.58) = 35

So, angle B = 35 inches.

Next, find the third angle.

120 + 35 + angle C = 180

angle C = 45

Next find side c using the Law of Sines.

15/sin(120) = c/sin(45)

c = 15/sin(120) x sin(45) = 12

side c = 12 inches