Question

find the value of x. 140° (5x)° x =

Explanation:

Step1: Analyze angle relations

The sum of angles around a point is \(360^\circ\), but here we can use linear pair and right angle. The angle adjacent to \(140^\circ\) (linear pair) and the right angle (\(90^\circ\)) and \(5x\) should sum to \(180^\circ\)? Wait, no. Wait, the straight line (linear pair) with \(140^\circ\) is \(180 - 140 = 40^\circ\)? Wait, no, looking at the diagram, \(DH\) is vertical, \(HF\) is horizontal (right angle, \(90^\circ\)). Wait, the angle between the line with \(140^\circ\) and \(DH\) and the angle between \(DH\) and \(HE\) and \(HF\)? Wait, maybe better: the angle opposite to \(140^\circ\) (vertical angles? No, wait, the angle adjacent to \(140^\circ\) (linear pair) is \(180 - 140 = 40^\circ\). Then, the right angle is \(90^\circ\), so \(40^\circ + 90^\circ + 5x = 180^\circ\)? Wait, no, the straight line (horizontal or vertical? Wait, \(DH\) is vertical, \(HF\) is horizontal, so they are perpendicular (right angle). The angle between the line \(HJ\) (with \(140^\circ\) adjacent) and \(DH\) is \(180 - 140 = 40^\circ\) (linear pair). Then, the angles at point \(H\): \(40^\circ\) (from linear pair) + \(90^\circ\) (right angle) + \(5x = 180^\circ\)? Wait, no, the sum of angles on a straight line (horizontal or vertical) is \(180^\circ\). Wait, the line \(DF\)? No, \(DH\) is vertical, \(HF\) is horizontal. So the angle between \(HJ\) (the line with \(140^\circ\) next to it) and \(DH\) is \(180 - 140 = 40^\circ\) (since they form a linear pair). Then, the angles \(40^\circ\), \(90^\circ\) (right angle between \(DH\) and \(HF\)), and \(5x\) should add up to \(180^\circ\) (since they are on a straight line? Wait, no, \(DH\) is vertical, \(HF\) is horizontal, so the line \(HE\) is between them. Wait, maybe the correct approach: the angle adjacent to \(140^\circ\) (linear pair) is \(40^\circ\) (because \(140 + 40 = 180\)). Then, the right angle is \(90^\circ\), so \(40 + 90 + 5x = 180\)? Wait, no, \(40 + 90 + 5x = 180\)? Let's check: \(40 + 90 = 130\), so \(130 + 5x = 180\), then \(5x = 50\), so \(x = 10\). Wait, that makes sense. Let's re-express:

The angle formed by the line with \(140^\circ\) and the vertical line \(DH\) is a linear pair, so it's \(180 - 140 = 40^\circ\). Then, the vertical line \(DH\) and horizontal line \(HF\) are perpendicular (right angle, \(90^\circ\)). So the three angles: \(40^\circ\) (from linear pair), \(90^\circ\) (right angle), and \(5x\) are on a straight line (horizontal? No, vertical? Wait, no, the horizontal line \(HF\) and the line \(HE\) and \(HJ\) related. Wait, maybe the correct equation is: the angle between \(HJ\) (the line with \(140^\circ\) adjacent) and \(DH\) is \(40^\circ\) (linear pair), then \(40^\circ + 90^\circ + 5x = 180^\circ\) (since they are on a straight line, the horizontal line? No, vertical? Wait, no, the sum of angles on a straight line is \(180^\circ\). So if we have a straight line (say, the line that is horizontal, \(HF\) and the extension), but maybe the correct way is:

Looking at the diagram, the angle marked \(140^\circ\), the right angle (\(90^\circ\)), and \(5x\) and the angle opposite? Wait, no, let's use the fact that the sum of angles around point \(H\) for the relevant lines: the angle adjacent to \(140^\circ\) (linear pair) is \(40^\circ\) (because \(140 + 40 = 180\)). Then, the right angle is \(90^\circ\), so \(40 + 90 + 5x = 180\) (since these three angles are on a straight line, forming a linear pair with the other side). So:

\(40 + 90 + 5x = 180\)

Step2: Solve for x

Simplify the left side: \(130 + 5x = 180\)

Subtract 130 from both sides: \(5x = 180 - 130 = 50\)

Divide both sides by 5: \(x = \frac{50}{5} = 10\)

Answer:

\(x = 10\)