Let’s look at a couple of examples of SAS. If two or more triangles have two equal sides with the same exact angle in between them, this means the given triangles are congruent. To avoid ambiguity you could instead have Step 2: Use Cosine Rule to find one of the other angles. SAS is true because the length of the third side is pre-determined if the length of the remaining two sides and the angle they form is known. What's clever about this choice is that there must be at least one angle greater than the angle you are finding, which guarantees that your angle can not be obtuse. Choosing to find the angle opposite the shorter of the two original sides guarantees that your angle is either the smallest angle in the triangle or is the second smallest (which might be the case if the original angle is the smallest angle). The Sine Rule is generally perceived as easier - there are fewer operations to perform, perhaps.Īnyway, having decided to opt for the Sine Rule as your method, we now have the possibility that you could have an ambiguous case, where it is unclear whether the angle you are finding should be acute or obtuse. You could either use the Cosine Rule to find one of the unknown angles or you could use the Sine Rule. Having followed Step 1, you now have three sides and one angle. The Sine Rule would be no help to you in that situation. To start with, you have no choice but to follow Step 1. You don't have to follow Step 2, but it helps. That way, we can use law of sines (which is easier than law of cosines), and we won't have to worry about getting more than one answer. So we should first try to find the angle that's opposite the shorter given side. Hence, we'll have just one answer for $\theta$: When you use the law of cosines to find some angle $\theta$ of a triangle, first you try to get $\cos\theta$ by itself, and you end up with an equation of the form Let's suppose $\theta$ is one of the angles, and we're trying to find $\theta$. At this point, we know all the sides of the triangle, we know one of the angles, and we're trying to find the other angles. Let's say that we've completed Step #1, and we're trying to decide what to do next.
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