In a triangle, the measure of the 2nd angle is twice the 1st angle. The 3rd angle is 15 degrees less than the 2nd angle. If there are 180 degrees in a triangle, find the measure of each angle.
The triangle is ABC, with associated angles α, β, γ.
Using the given information
the 2nd angle is twice the 1st angle -> β = 2α
The 3rd angle is 15 degrees less than the 2nd angle -> γ = β - 15
there are 180 degrees in a triangle -> α + β + γ = 180
Our three simultaneous eqns then are,
β = 2α --------------------- (1)
γ = β - 15 ------------------ (2)
α + β + γ = 180 ---------- (3)
Substituting for (1) into (2) and (3),
γ = 2α - 15 ------------------ (4)
α + 2α + γ = 180 ---------- (5)
rearranging,
γ - 2α = -15 ----------------- (6)
γ + 3α = 180 ---------------- (7)
Subtracting (6) from (7),
5α = 195
α = 39
β = 78
γ = 63
The solution then is: α = 39, β = 78, γ = 63