Question

1. in the figure, there are lines and angles: angle at c is 142°, angle at b is 5x°, angle at a is (4x - 2)°. points f, b; e, a; c, d are on lines respectively.

Explanation:

Step1: Find the supplementary angle at C

The angle adjacent to \(142^\circ\) on a straight line is \(180^\circ - 142^\circ = 38^\circ\). But wait, actually, in the triangle, we can use the exterior angle theorem or the fact that \(AB\) and \(CD\) might be related, but first, let's look at the angles. Wait, the angle at \(C\) adjacent to \(142^\circ\) is \(180 - 142 = 38^\circ\)? No, maybe better to use the fact that \( \angle BCD = 180 - 142 = 38^\circ\)? Wait, no, let's see the triangle \(ABC\). Wait, actually, the angle at \(A\) is \((4x - 2)^\circ\), angle at \(B\) is \(5x^\circ\), and angle at \(C\) should be equal to \(180 - 142 = 38^\circ\) (since they are supplementary). Wait, no, maybe the exterior angle or the sum of angles in a triangle. Wait, the sum of angles in a triangle is \(180^\circ\), so \(5x + (4x - 2) + (180 - 142) = 180\)? Wait, no, the angle at \(C\) is equal to \(180 - 142 = 38^\circ\) because they are linear pairs. So then, in triangle \(ABC\), the sum of angles is \(5x + (4x - 2) + 38 = 180\)? Wait, no, maybe the angle at \(C\) is equal to \(180 - 142 = 38^\circ\), so:

\(5x + (4x - 2) + 38 = 180\)? Wait, no, that would be if it's a triangle, but maybe the lines \(AB\) and \(CD\) are parallel? Wait, the diagram shows \(FB\) and \(EA\) as parallel? Wait, maybe the angle at \(C\) is equal to the angle at \(A\) if they are alternate interior angles? Wait, no, let's re-examine.

Wait, the angle at \(C\) (the one inside the triangle) is \(180^\circ - 142^\circ = 38^\circ\)? No, that's not right. Wait, the straight line \(CD\) and the transversal, so the angle adjacent to \(142^\circ\) is \(180 - 142 = 38^\circ\), so that angle is \(38^\circ\). Then, in the triangle \(ABC\), the sum of angles is \(5x + (4x - 2) + 38 = 180\)? Wait, no, maybe the angle at \(C\) is equal to \((4x - 2)^\circ\)? No, that doesn't make sense. Wait, maybe the angle at \(C\) is equal to \(180 - 142 = 38^\circ\), and then the sum of angles in the triangle is \(5x + (4x - 2) + 38 = 180\). Let's solve that:

\(5x + 4x - 2 + 38 = 180\)

\(9x + 36 = 180\)

\(9x = 144\)

\(x = 16\)

Wait, but let's check again. Alternatively, maybe the angle at \(C\) is equal to \((4x - 2)^\circ\) because of vertical angles or alternate interior angles. Wait, the line \(AD\) is a straight line, so angle at \(A\) is \((4x - 2)^\circ\), angle at \(C\) is \(180 - 142 = 38^\circ\), and angle at \(B\) is \(5x^\circ\). Then sum of angles in triangle: \(5x + (4x - 2) + 38 = 180\)

\(9x + 36 = 180\)

\(9x = 144\)

\(x = 16\)

Wait, but let's verify. If \(x = 16\), then angle at \(B\) is \(5*16 = 80^\circ\), angle at \(A\) is \(4*16 - 2 = 62^\circ\), angle at \(C\) is \(38^\circ\). Sum: \(80 + 62 + 38 = 180\), which works. Alternatively, maybe the angle at \(C\) is equal to \((4x - 2)^\circ\) and angle at \(B\) is \(5x^\circ\), and the angle at \(C\) is supplementary to \(142^\circ\), so \( (4x - 2) + 5x + (180 - 142) = 180\)? Wait, no, that's the same as before. So solving \(9x + 36 = 180\) gives \(x = 16\).

Step2: Solve for x

Wait, let's do the equation again. Sum of angles in triangle: \(5x + (4x - 2) + (180 - 142) = 180\)

Simplify: \(5x + 4x - 2 + 38 = 180\)

\(9x + 36 = 180\)

Subtract 36: \(9x = 144\)

Divide by 9: \(x = 16\)

Answer:

\(x = 16\)