Answer:
x = 2.75+√30.5625
∠3 = ∠5 ≈ 113.923°
Explanation:
We are given that ∠3 = (x+1)(x+4) and ∠5 = 16(x+3)-(x²-2) are corresponding angles, hence equal. We can equate the two angle expressions and solve the resulting quadratic for x.
... (x+1)(x+4) = 16(x+3)-(x²-2)
... x² +5x +4 -16x -48 +x² -2 = 0 . . . . . subtract the right side, eliminate parentheses
... 2x² -11x -46 = 0 . . . . . . . . . . . . . . . . . collect terms
Using the quadratic formula, we want to find
... x = (-b±√(b²-4ac))/(2a) . . . . for a=2, b=-11, c=-46
... x = (11 ±√((-11)² -4(2)(-46)))/(2(2)) = (11 ±√489)/4 = 2.75 ± √30.5625
The negative solution results in negative values for the angles, so only the positive solution is useful for this problem.
... x = 2.75+√30.5625 ≈ 8.27834
Using this value for x in either expression for the angle value, we get
... ∠3 = ∠5 = (8.27834+1)(8.27834+4) ≈ 113.923 . . . degrees
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It seems a little odd that this problem should result in irrational values for the variables. If we take ∠3 and ∠5 to be a linear pair, then the solution is x=6 and the angle measures are 70° and 110°. The solution is done basically the same way, except that you use the equation
... ∠3 + ∠5 = 180
and substitute the given expressions. The x² terms will cancel, leaving a linear equation easily solved.
(Since this is not the problem described here, the detailed working is left to the reader.)