Given:
m∠C = 85°, a= 10, c = 13
From the Law of Sines,
sin(A)/a = sin(C)/c
sin(A) = (a/c)*sin(C)
= (10/13)*sin(85°)
= 0.7663
m∠A = sin⁻¹ 0.7663 = 50°, or 130°
When m∠A = 50°, obtain
m∠B = 180° - (m∠A + m∠C) = 180 - (50+85) = 45°
Again from the Law of Sines, obtain
b = (sinB/sinC)*c = 9.2
When m∠A = 130°, obtain
m∠B = 180° - (130 + 85) = -35° (not possible)
Therefore this triangle does not exist.
Answer:
There is only one possible triangle, with
A=50°, B=45°, C=85°, a=10, b=9.2, c=13.