When is there an ambiguous case

If A is obtuse and a b , no triangle exists. Situation 2: TWO different triangles exist. We can see right away that the problem that existed in Example 1 is not a problem with this triangle.

The value of a is larger than the height from C 8 and a triangle will be formed. But, if we swing side a from point C to the left, can we form a second triangle?

Looks good! But wait! We know that the sine of an obtuse angle is the sine of its supplement. But since a is smaller than b , it can "swing" to the left of h and create a second triangle containing an obtuse angle.

When the angle is acute, five possible solutions exist. When the angle is obtuse, three possible solutions exist. Let a , b , and B be known, and let B be acute. Five different cases exist. Exactly one solution exists, and a right triangle is determined. Each of these five case is illustrated below. Let a , b , and B be known, and let B be obtuse. Now, consider a case where angle A is 58 degrees, side a is 25 units long, and side b is 22 units long. Using this method, we find that the one triangle exists with an angle B equal to 48 degrees.

Does another exist? To prove that another triangle exists, we must determine if another possible angle B exists that makes the triangle possible. Notice the isosceles triangle created by the Ambiguous Case above. This means the second angle B is degrees. Since we know all angles in the triangle must add to degrees, a second triangle is not possible because angles A and B alone exceed degrees. If a second triangle was possible, angles A and B would sum to a number smaller than degrees.

You should find that angle B is roughly 69 degrees. This time angle A and angle B sum to less than degrees, making an angle C possible it measures roughly 11 degrees. Therefore, a second triangle is possible. High School. By Maryam Amr. The Law of Sines The Law of Sines relates all angles and sides of a triangle in the following way, in which the lowercase letters indicate the side directly across from the capitalized angle: A Refresher: Showing Congruence By now, you know that you can use a combination of sides and angles to prove congruence between two triangles.

Example 2 - One Triangle Exists Now, consider a case where angle A is 58 degrees, side a is 25 units long, and side b is 22 units long. Health Professions.