# The Ambiguous Case

## Key Questions

As listed below.

#### Explanation:

For those of you who need a reminder, the ambiguous case occurs when one uses the law of sines to determine missing measures of a triangle when given two sides and an angle opposite one of those angles (SSA). ... If angle A is acute, and a = h, one possible triangle exists

1. If angle A is acute, and a < h, no such triangle exists.

2. If angle A is acute, and a = h, one possible triangle exists.

3. If angle A is acute, and a > b, one possible triangle exists.

4. If angle A is acute, and h < a < b, two possible triangles exist.

5. If angle A is obtuse, and a < b or a = b, no such triangle exists.

6. If angle A is obtuse, and a > b, one such triangle exists.

• Three numbers ($a , b , c$) can be the lenght of three sides of a triangle if and only if each of them is greater then the difference of the other two, and less of the sum of the other two.

I.E: (if $a > b > c$)

$a > b - c$,
$b > a - c$,
$c > a - b$;

and:

$a < b + c$,
$b < a + c$,
$c < a + b$.

As listed below.

#### Explanation:

If the sum is over 180°, then the second angle is not valid. First we know that this triangle is a candidate for the ambiguous case since we are given two sides and an angle not in between them. We need to find the measure of angle B using the Law of Sines: If their sum is less than 180°, we know a triangle can exist.

To determine if there is a 2nd valid angle:

1. See if you are given two sides and the angle not in between (SSA). This is the situation that may have 2 possible answers.

2. Find the value of the unknown angle.

3. Once you find the value of your angle, subtract it from 180° to find the possible second angle.

4. Add the new angle to the original angle. If their sum is less than 180°, you have two valid answers. If the sum is over 180°, then the second angle is not valid.

5. If already one obtuse angle given, it can not have a second set of values.

• To solve a triangle it is necessary to know at least three elements with an only exception: if these three elements are the three angles.

In fact if two triangles have identical the three angles, they are similar.