The graph above shows why we can sometimes have no possible triangles sometimes one and sometimes two. Adjust the sliders for [math]a[/math], [math]b[/math] and [math]\theta[/math]. You can also drag around point [math]B[/math], but it will always be on the circle centered at [math]C[/math].[br][br]If we are given two sides of a triangle and an angle that is not between them (SSA):[br][br][b][u]Method 1: Check both Angles and Reject if Necessary[/u][/b][br][list=1][*]Use the Law of Sines to find sine of the angle[/*][*]Find both angles in Quadrant I and II with the corresponding reference angle.[/*][*]Find the third angle of the triangle[/*][*]Reject any impossible triangle.[/*][/list][br][b][u]Method 2: Check [/u][/b][math]h[/math][u], the altitude from [/u][math]C[/math][u] to the opposite side[br][/u][br][b]Case I: [/b][math]\angle A[/math][b] is acute[/b][br]Suppose we let [math]m\angle A[/math] and [math]a[/math] have fixed values and we just adjust the length of [math]b[/math]. For example, let [math]m\angle A=30^\circ[/math] and [math]a=5[/math]. If we turn on the circle where [math]B[/math] must lie, we can see that changing the length of [math]B[/math] will make the circle intersect the opposite side once, twice, or not at all.[br][br]Note that since [math]\sin\angle A=\frac{h}{b}[/math], we can express the altitude as: [math]h=b\sin\angle A[/math]
If [math]h>a[/math], how many triangles can be formed?
None, because [math]a[/math] will be too short to reach the opposite side. The circle and opposite side do not intersect.
If [math]h=a[/math], how many triangles can be formed?
One, because [math]a[/math] will be just long enough to reach the opposite side. The circle will be tangent to the opposite side, meeting it at exactly one point. It will form a right angle at [math]B[/math].
If [math]a\geq b[/math], how many triangles can be formed?
One, because the circle will intersect the opposite side in just one point. If [math]a=b[/math], the the triangle is isosceles and a second triangle cannot be formed. If [math]a>b[/math], then the circle extends beyond [math]A[/math] and will not meet the opposite side a second time.
[b]Case II: [/b][math]\angle A[/math][b] is right or obtuse[/b]
If [math]a\leq h[/math], how many triangles can be formed?
Zero, because the circle will not meet the opposite side or it will be tangent to it, but that would result in just a line segment, not a triangle.
If [math]a>h[/math], how many triangles can be formed?
One. It will be a right triangle. The second angle for [math]\angle B[/math] cannot be possible, as it would be obtuse, but you cannot have a right/obtuse angle as well as an obtuse angle in the same triangle.