In the given figure, PQ=PR and QS is the bisector of angle PQR. If angle PQS=30°, find angle PSQ.

90°

Step-by-step explanation:

PQ=PR ⇒

∠PQR = ∠PRQ

QS is the bisector of angle PQR ⇒

∠PQS= ∠RQS

If angle PQS=30°, find angle PSQ

∠PQS=30° ⇒

∠PQR = 2*∠PQS = 2*30° = 60°

As ∠PQR = 60°,

∠PRQ = 60° and ∠QPR = 60° as remaining angle

So,

∠PSQ = 180° - (∠QPR + ∠PQS) = 180° - 90° = 90°

When given 3 triangle sides, to determine if the triangle is acute, right or obtuse:
1) Square all 3 sides.
2) Sum the squares of the 2 shortest sides.
3) Compare this sum to the square of the 3rd side.

if sum > 3rd side²   Acute Triangle
if sum = 3rd side²   Right Triangle
if sum < 3rd side²   Obtuse Triangle

1) 1,296   2,401   3,600
2) Sum = 3,697
3) 3,697 is greater than 3,600
Therefore, the triangle is acute.

Source:
http://www.1728.org/triantest.htm