Question

How you know when there is only one solution triangle possible from a side-side-angle?

Answer 1

There will be only on solution if either of these conditions is met:

• the given angle is opposite the longest side
• the triangle is a right triangle (the ratio of the side opposite the angle to the other given side is equal to the sine of the angle)

Explanation:

Consider the SSA geometry with the angle placed in standard position at the origin and its adjacent given side (the second side of SSA) extending along the +x axis. Draw a circle having a radius equal to the length of the first side of SSA, centered on the vertex between the two segments on the +x axis.

There are three possibilities for the way this circle intersects the other ray of the angle (the ray that is not the x-axis):

1. the first side is as long or longer than the second side, so there is one point of intersection (one solution to the triangle, purple in the attachment)
2. the first side is just long enough to be tangent to the other ray, so there is one point of intersection (one solution to the triangle, green in the attachment)
3. the first side is between these lengths, so intersects the other ray in two places, giving two solutions to the triangle, red in the attachment.