a The triangle is isosceles which means it has a pair of congruent legs and a pair of congruent base angles. We know that one of these base angles is 52^(∘). With this we can find the measure of the vertex angle.
θ + 52^(∘) + 52^(∘) = 180^(∘) ⇔ θ = 76^(∘)
Let's add this information to the diagram.
Now we have enough information to solve for x with the Law of Sines.
c Since we have been given two angles in this triangle, we can find the measure of the last angle by equating the sum of the known angle measures with 180^(∘) and solving for the unknown angle.
θ +5^(∘)+85^(∘) = 180^(∘) ⇔ θ = 90^(∘)
With the extra piece of information, we can use the Law of Sines to determine x.
Finally, to find the solutions, we have to split the fraction into the positive and negative case.
Positive:& - 7 + 17/2= 5 [0.5em]
Negative:& - 7 - 17/2=- 12
Note that a side in a triangle cannot be negative making x=5 the only viable solution.
