Question

For each of the following, tell how many non congruent triangles PQR fit the given description, and find the size of angle Q. Make a separate diagram for each case. (a) p = 3, q = 5, angle P = 27 degrees (b) p = 8, q = 5, angle P = 57 degrees (c) p = 7, q = 8, angle P = 70 degrees (d) p = 10, q = 20, angle P = 30 degrees.

Answer 1

consider a right triangle PQR, with Q=90°. p = q sinP
So, in any other triangle, if given p,q, and P
if p < q sinP, then it is too short to meet side r, so no triangle possible
If p > q sinP, then there will be two triangles, one obtuse and one acute.

So, for problem a above, since 3 > 5 sin27°, there are two triangles that work

See what you can do with the others.