ASA Triangles

The letters ASA stand for angle – side – angle, meaning a triangle with two known angles and a known length of one side between the two angles. When we have this much information, we can figure out the remaining angle and the length of the other two sides.

Finding The Third Angle Of An ASA Triangle

Imagine a triangle with sides a, b, and c, where the length of a = 5 inches. We know that the angle where a and b meet is 60 degrees. We know that the angle where a and c meet is 90 degrees. To find out the third angle, we only need to remember the rule that the sum of the three angles of any triangle is always 180. In this example, 60 + 90 + ? = 180. Since 60 + 90 = 150, and 180 – 150 = 30, we know that the angle where b meets c is 30 degrees.

Finding The Two Remaining Sides Of An ASA Triangle

The formula for finding the sides of an ASA triangle is a little more complex, and it requires the sine function on a calculator. First let’s picture a triangle like the one above with sides a, b, and c. Let’s denote the angle opposite each side with a capital letter, so that the angle opposite side a is A, the angle opposite side b is B, and the angle opposite side c is C. Using these designations, the Law of Sines states that

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

Therefore, as long as we know at least two angles and one opposite side, we can use the formula to find the lengths of the other two sides.

In the example given above, we know that one side (a) has a length of 5 inches, and we know that the opposite angle A (where b meets c) is 30 degrees. We also know that angle B opposite side b is 90 degrees. We can use this information to set up the equation: 5/sin(30) = b/sin(90).

Using algebra, we can rearrange the equation as

b = 5/sin(30) x sin(90)

b = 5/.5 x 1

b = 10 inches

We can then do the same thing with side c, knowing that its opposite angle C is 60 degrees. We can use either a/sin(A) or b/sin(B) to calculate c. Using the first side and angle, a/sin(A), we can set up the new equation as 5/sin(30) = c/sin(60). By rearranging the equation, we can work it out through the following steps.

c = 5/sin(30) x sin(60)

c = 5/.5 x .87

c = 8.7 inches

Practice

Problem: Given the following information about an ASA triangle, solve for angle C and sides a and b.

Angle A = 50 degrees

Angle B = 70 degrees

side c = 6 inches

Solution:

To find the third angle, add A and B and subtract from 180. Angle C is 60 degrees.

To find side a, use the sine rule: a/sin(50) = 6/sin(60)

a = 6/sin(60) x sin(50)

a = 6/.87 x .77

a = 5.3 inches

To find side b, use the sine rule b/sin(70) = 6/sin(60)

b = 6/sin(60) x sin(70)

b = 6/.87 x .94

b = 6.5 inches