Determine the value of X in the following diagram

Question

I have provided the link for the diagram:

https://ibb.co/0Vpqxs5

Answer 1

Let

α be the angle to the left of the 68° angle,

β be the angle to the right of the 68° angle,

Y be the remaining angle in the triangle with angles β, X, Y

v be the perpendicular distance from the vertex of Y to the line connecting β and X.

Then we have the following four identities with four unknowns

α + β + 68° = 180° (1)

tan(α) = 15/25 (2)

sin(β) = v/25 (3)

sin(X) = v/35 (4)

From (2)

α = 30.96375653°

From (1)

β = 81.03625347°

From (3)

v = 25 sin(β) = 24.69467746

From (4)

sin(X) = 0.705562213

X = 44.87498231°

X = 135.1250177°

The problem has two solution.

The solution of about 45° is the one suggested in the picture.

The other solution of about 135° is also a perfectly valid solution,

if we were not required to interpret the picture as being to scale.

Answer 2

In right triangle angle Y opposite to 15 cm leg: tanY = 15cm/25cm = 0.6, so Y = tan^-10.6 = 31°.

Then angle Z opposite to side 35 cm is 180° - (68° + 31°) = 81°.

By Sine Law: sinX/25 = sin81°/35°; sinX = 25sin81°/35 = 0.7055; X = sin^-10.7055 = 44.87° ≈ 45°.