# How do you determine the number of possible triangles and find the measure of the three angles given AB=14, BC=21, mangleC=75?

Nov 1, 2017

No Triangle

#### Explanation:

Since the given information is for a SSA triangle it is the ambiguous case. In the ambiguous case we first find the height by using the formula $h = b \sin A$.

Note that A is the given angle and its side is always a so the other side will be b .

So if $A < {90}^{\circ}$ and if

1. $h < a < b$ then then there are two solutions or two triangles.

2. $h < b < a$ then there is one solution or one triangle.

3. $a < h < b$ then there is no solution or no triangle.

If $A \ge {90}^{\circ}$ and if

1. $a > b$ then there is one solution or one triangle.

2. $a \le b$ there is no solution

$h = 21 \sin {75}^{\circ} \approx 20.28$, since $14 < 20.28 < 21$ we have

$a < h < b$ so we have no solution or no triangle.