What is an SSA Triangle?

Definition

An SSA triangle is a triangle where we know the length of two sides and one angle. SSA stands for side, side, angle, and we can use those known variables to determine the unknown side and angles of a given triangle. So, an SSA triangle is when:

  • We have two known side lengths of the triangle.
  • We are given an angle measurement which is not in between the two sides given.

After determining that it is an SSA triangle upon checking the checklist above, we can now determine the other unknown side and the remaining unknown two angles.

Example of an SSA Triangle

Look at the illustration below, we have a triangle with sides a, b, and c and angles A, B, and C, respectively. This is an example of an SSA Triangle. Take a closer look at the variables used to denote the three angles and three sides of the given triangle. It is important to know the distinction of these variables to avoid any error in your future calculation.

Based on the triangle given, the length of sides a and b are given. The length of side c is still unknown.

  • a = 9 mm
  • b = 5 mm

One angle measurement is also given which is angle A. The two angles B and C are not given.

  • A = 63 °

Back to the above definition of an SSA Triangle, we can first evaluate if it satisfies the two key points that we have mentioned.

  • Two side lengths of the triangle are given. As we can see from the example above, the length of the two sides of the triangle is given which is a = 9 mm and b = 5 mm.
  • One angle measurement is given. In the case above, measurement of angle A is given which is A = 63°

The two key points are present so we can conclude that the given triangle is an SSA Triangle.

SSA Triangle Application

Now we can now use the proper approach to find the remaining unknown side and angles. The very reason why we need to first determine what kind of triangle is given is that when we know that it is an SSA Triangle, we can automatically use Sine Law in finding the remaining unknown quantities.

Let’s have a short review of what Sine Law is.

Sine Law is an equation that relates the sides of the triangle to its corresponding angles. Also, make sure you already know the basics of Algebra when using this formula. We will need basic Algebra in manipulating this equation.

Sine Law

The Law of Sines simply states that the ratios below are equal:

\dfrac{a}{sin(A)} = \dfrac{b}{sin(B)} = \dfrac{c}{sin(C)}

Please take note that the small letters a, b, and c correspond to the length of the sides of the triangle. While the capital letters, A, B, and C correspond to the angles of the triangle.

Using the above example illustration, let us follow the steps below to get the remaining length and angles that are not given.

First, write the given side lengths and known angle of the triangle in the above example. This will help us to easily input these values to our computation in the next steps.

a = 9 mm

b = 5 mm

A = 63 °

Find the unknown angle B using Sine Law. We choose to determine angle B first than angle C since side b is already known. In selecting what angle is to be determined first, we must be reminded that there should be one unknown variable in the equation we choose to evaluate which in this case is angle B.

\dfrac{a}{sin(A)} = \dfrac{b}{sin(B)}

\dfrac{9}{sin(63)} = \dfrac{5}{sin(B)}

10.10 = \dfrac{5}{B}

B = \dfrac{5}{10.10}

B = sin^{-1} \Big(\dfrac{5}{10.10}\Big)

B = 29.67\degree

Note: The angle B in the above calculation is rounded to only 2 decimal places. You can add more decimal places as required.

Now, we already have two known angles of the triangle. We will use these two known angles to determine the third angle which is angle C. We know that the total angle measurement of a triangle is 180 degrees. So:

A + B + C = 180

Angle A and angle B are already known so let’s substitute the two angles in the equation above to find angle C. We still need basic algebra in this step. Please take note how the terms are transposed to the right side of the equation.

63 + 29.67 + C = 180

C = 180 - 63 - 29.67

C = 87.33\degree

Note: The final answer in the above calculation is rounded to only 2 decimal places since we use the rounded angle B value. You can add more decimal places as required.

Now we know the measurement of the three angles of the triangle, we can solve for the remaining side c. In this step, we can use two ways to obtain the length of side c.

  1. Using the given lenth of side a and angles A and C
  2. Using the calculated angles B and C and the given length of side b

Method 1

Using the given length of side a and angles A and C is the safest choice since two of the variables (side a and angle A) are from the given. Only angle C is from your calculation which means you have a little chance to get a wrong answer unless your calculated angle C is incorrect in the first place.

Substitute the known length of side a and the angles A and C in the Sine equation:

\dfrac{a}{sin(A)} = \dfrac{c}{sin(C)}

\dfrac{9}{63} = \dfrac{c}{sin(87.33)}

10.10 = \dfrac{c}{0.9989144}

c = (10.10) \times (0.9989144)

c = 10.09mm

Note: The final answer in the above calculation is rounded to only 2 decimal places. You can add more decimal places as required.

Method 2

Using our calculated angles B and C and the given length of side b is risky if your calculated angles B and C in the previous steps are incorrect. This will possibly lead to an incorrect answer for the length of side c. However, if your calculated angles B and C in the previous steps are correct, you don’t need to worry. This approach will still lead you to the same correct answer.

Substitute your calculated angles B and C and the given length of side b in the Sine equation. Still, Algebra is still needed in this part.

\dfrac{b}{sin(B)} = \dfrac{c}{sin(C)}

\dfrac{5}{sin(29.67)} = \dfrac{c}{sin(87.33)}

10.10 = \dfrac{c}{0.9989144}

c = (10.10) \times (0.9989144)

c = 10.09mm

Look how we arrived at the same answer with the first option. You can use these two approaches for error checking of your first answer. If the two arrived with the same answer, then you made a good computation. However, if the two arrived with different values, then, there should be mistake in your calculation. Review everything and do it carefully and correctly.

Ambiguous Cases

SSA triangles can be tricky. There are things we need to know when there is ambiguity. I will share two possible scenarios when you are dealing with SSA triangle ambiguity.

No Triangle

Even when the two key points mentioned in identifying an SSA triangle are present, it can happen that no triangle exists from the given side lengths and angles. This happens when you are given lengths of sides a and b and angle A and the following conditions are present:

  • A > 90
  • a<b
  • sin B > 1

Two Triangles

Sometimes, we directly assume that we already get the values of the unknown side and angles after solving directly. However, it is also possible that we only solved the first triangle only. In SSA triangle, there are times when two triangles can be made from the given sides and angle.

This is a case when you are given lengths of sides a and b and angle A and the following conditions are present:

  • A < 90
  • a<b

In this case, two triangles can be formed and are located in Quadrants I and II.

Let’s have an example of ambiguous case with two triangles where A = 40°, a = 2, and b = 3.

Let’s first check if it meets the two triangle conditions we stated above.

  • A < 90 – this is true because A = 40
  • a < b – this is also true because a is 2, which is less than 3.

Let’s first solve for the first triangle unknown side and angles. Let’s put a subscript 1 to denote that it belongs to the first triangle that we will solve.

\dfrac{a}{sin(A)} = \dfrac{b}{sin(B)_{1}}

\dfrac{2}{sin(40)} = \dfrac{3}{sin(B)_{1}}

3.11 = \dfrac{3}{B_{1}}

B_{1} = \dfrac{3}{3.11}

B_{1} = sin^{-1} \Big(\dfrac{3}{3.11}\Big)

B_{1} = 74.62\degree

A + B_{1} + C_{1} = 180

40 + 74.62 + C_{1} = 180

C_{1} = 180 - 40 - 74.62

C_{1} = 65.38\degree

\dfrac{a}{sin(A)} = \dfrac{c_{1}}{sin(C)_{1}}

\dfrac{2}{sin(40)} = \dfrac{c_{1}}{sin(65.38)}

3.11 = \dfrac{c_{1}}{0.9090907442}

c_{1} = (3.11) \times (0.9090907442)

c_{1} = 2.83

Now, let’s solve for the other triangle that is possible. Let’s put a subscript 2 to the unknown side and angles to denote that it belongs to the second triangle that we will solve.

Based on the given variables, we are certain that it is an ambiguous case with two triangles, and so we can directly solve for B2 by subtracting angle B1 to 180 directly.

B_{2} = 180 - B_{1}

B_{2} = 180 - 74.62

B_{2} = 105.38

A + B_{2} + C_{2} = 180

40 + 105.38 + C_{2} = 180

C_{2} = 180 - 40 - 105.38

C_{2} = 34.62\degree

\dfrac{a}{sin(A)} = \dfrac{c_{2}}{sin(C)_{2}}

\dfrac{2}{sin(40)} = \dfrac{c_{2}}{sin(34.62)}

\dfrac{2}3.11 = \dfrac{c_{2}}{0.5681310392}

c_{2} = (3.11) \times (0.5681310392)

c_{2} = 1.77

Conclusion

Hopefully, this article has helped you to understand what SSA triangles are and how you can work with them, calculate them, and handle and ambiguous cases that you may come across.