Question

find the value of x. (image of angle diagram with points h, b, c, d, e, f, g; angles 32°, 32°, (x + 8)° and a right angle at h for e and b related lines)

Explanation:

Step1: Analyze angle relationships

We know that the angle between \(HB\) and \(HE\) is \(90^\circ\) (right angle). The sum of angles around a point or in a straight - line (but here we use the right - angle and given angles) should help us find the relationship for the angle \((x + 8)^\circ\). First, let's find the angle between \(HD\) and \(HB\). The angle between \(HB\) and \(HC\) is \(32^\circ\), and the angle between \(HC\) and \(HD\) is such that the angle between \(HD\) and \(HE\) is \(32^\circ\). So the angle between \(HB\) and \(HD\) is \(32^\circ+32^\circ = 64^\circ\), and since the angle between \(HB\) and \(HE\) is \(90^\circ\), the angle between \(HD\) and \(HE\) is \(90^\circ-(32^\circ + 32^\circ)=26^\circ\)? Wait, no, actually, looking at the vertical - angle or the angle that is equal to the angle between \(HB\) and \(HC\) and the other angle. Wait, the angle \((x + 8)^\circ\) should be equal to the angle between \(HB\) and \(HD\) minus the right - angle? No, let's re - examine.

Wait, the key is that the angle \((x + 8)^\circ\) and the angle formed by \(32^\circ+32^\circ\) should add up to \(90^\circ\)? Wait, no. Let's see: the angle between \(HB\) and \(HE\) is \(90^\circ\). The angle between \(HB\) and \(HC\) is \(32^\circ\), the angle between \(HC\) and \(HD\) is some angle, and the angle between \(HD\) and \(HE\) is \(32^\circ\). Also, the angle \((x + 8)^\circ\) is vertically opposite or equal to the angle between \(HB\) and \(HD\) minus the right - angle? Wait, no. Let's use the fact that the sum of angles in a right - angled situation. Wait, the angle \((x + 8)^\circ\) plus \(32^\circ+32^\circ\) should equal \(90^\circ\)? Wait, no, let's think again.

Wait, the angle between \(HB\) and \(HE\) is \(90^\circ\). The angle between \(HB\) and \(HC\) is \(32^\circ\), the angle between \(HC\) and \(HD\) is such that the angle between \(HD\) and \(HE\) is \(32^\circ\). So the angle between \(HB\) and \(HD\) is \(32^\circ + 32^\circ=64^\circ\), and the angle between \(HD\) and \(HE\) is \(90^\circ - 64^\circ = 26^\circ\)? No, that's not right. Wait, actually, the angle \((x + 8)^\circ\) is equal to \(90^\circ-(32^\circ + 32^\circ)\)? Wait, no, let's set up the equation.

We know that the angle \((x + 8)^\circ\) and the angle \(32^\circ+32^\circ\) should add up to \(90^\circ\) because of the right angle at \(H\) (between \(HB\) and \(HE\)). So:

\((x + 8)+32 + 32=90\)

Step2: Solve the equation for \(x\)

First, simplify the left - hand side of the equation:

\(x+8 + 32+32=x + 72\)

So we have the equation \(x + 72=90\)

Subtract \(72\) from both sides:

\(x=90 - 72\)

\(x = 18\)

Wait, let's check again. The angle \((x + 8)^\circ\) and the two \(32^\circ\) angles are in the right - angle (since the angle between \(HB\) and \(HE\) is \(90^\circ\)). So the sum of \((x + 8)^\circ\), \(32^\circ\), and \(32^\circ\) is \(90^\circ\). So:

\(x + 8+32 + 32=90\)

\(x+72 = 90\)

\(x=90 - 72=18\)

Answer:

\(x = 18\)