Congruence Criteria Math Example 4

Solution

1. 1
   Step 1: In SSA, you know two sides and the angle opposite one of them (not between the two sides). Given side $a$ (opposite the known angle $\theta$), side $b$, and angle $\theta$, try to construct the triangle.
2. 2
   Step 2: Fix side $b$ and angle $\theta$. The third vertex lies on a circle of radius $a$ centered at the end of $b$. This circle can intersect the opposite ray in 0, 1, or 2 points — two points means two different valid triangles with the same SSA data.
3. 3
   Step 3: Example: $b = 10$, $a = 8$, $\theta = 35°$. There can be two distinct triangles (the 'ambiguous case'). Therefore SSA does not guarantee a unique triangle, so it is not a valid congruence criterion.