Question

50)
s r
t u
24x - 1 85°

Explanation:

Step1: Identify the figure type

The figure is a rectangle (since \(STUR\) has right angles implied, and adjacent angles in a rectangle are supplementary? Wait, no—wait, in a rectangle, all angles are \(90^\circ\)? Wait, no, maybe it's a parallelogram? Wait, no, the diagram shows \(T\) and \(U\) with angles \(24x - 1\) and \(85^\circ\). Wait, in a rectangle, adjacent angles are equal? No, wait, in a rectangle, all angles are \(90^\circ\), but maybe this is a parallelogram where consecutive angles are supplementary? Wait, no, maybe it's a rectangle, so angles at \(T\) and \(U\) are equal? Wait, no, in a rectangle, all angles are \(90^\circ\), but here angle at \(U\) is \(85^\circ\)? Wait, maybe it's a parallelogram, and consecutive angles are supplementary? Wait, no, maybe the figure is a rectangle, so angles \(T\) and \(U\) are equal? Wait, no, maybe I made a mistake. Wait, the figure is labeled \(S\), \(R\), \(U\), \(T\) as a quadrilateral, probably a rectangle, so angles at \(T\) and \(U\) are right angles? Wait, no, the angle at \(U\) is \(85^\circ\), and angle at \(T\) is \(24x - 1\). Wait, maybe it's a parallelogram, so consecutive angles are supplementary? No, in a parallelogram, consecutive angles are supplementary, but if it's a rectangle, all angles are \(90^\circ\). Wait, maybe the figure is a rectangle, so angle \(T\) and angle \(U\) are equal? Wait, no, that can't be. Wait, maybe the diagram is a rectangle, so angle \(T\) is equal to angle \(U\)? Wait, no, in a rectangle, all angles are \(90^\circ\), so \(24x - 1 = 85\)? Wait, that would make \(24x = 86\), \(x = 86/24 = 43/12 \approx 3.58\), but that doesn't make sense. Wait, maybe it's a parallelogram, and angle \(T\) and angle \(U\) are equal? No, consecutive angles in a parallelogram are supplementary. Wait, maybe the figure is a rectangle, so angle \(T\) is \(90^\circ\), but angle \(U\) is \(85^\circ\)? That can't be. Wait, maybe the diagram is a rectangle, so angle \(T\) and angle \(U\) are adjacent angles, so they should be supplementary? No, in a rectangle, adjacent angles are \(90^\circ\), so they are equal and supplementary (since \(90 + 90 = 180\)). Wait, maybe the problem is that it's a rectangle, so angle \(T\) equals angle \(U\)? So \(24x - 1 = 85\). Let's solve that.

Step2: Solve for \(x\)

Set \(24x - 1 = 85\) (assuming angles at \(T\) and \(U\) are equal, as in a rectangle, opposite angles are equal, and consecutive angles are supplementary, but if it's a rectangle, all angles are \(90^\circ\), but maybe the diagram is a parallelogram with \(T\) and \(U\) as adjacent angles? No, in a parallelogram, adjacent angles are supplementary. Wait, maybe the problem is that it's a rectangle, so angle \(T\) is \(90^\circ\), but the given angle at \(U\) is \(85^\circ\)? That's conflicting. Wait, maybe the figure is a rectangle, so angle \(T\) and angle \(U\) are equal, so \(24x - 1 = 85\). Then:

\(24x - 1 = 85\)

Add 1 to both sides:

\(24x = 86\)

Divide both sides by 24:

\(x = \frac{86}{24} = \frac{43}{12} \approx 3.58\). But that seems odd. Wait, maybe the figure is a parallelogram, and angle \(T\) and angle \(U\) are supplementary? So \(24x - 1 + 85 = 180\). Let's try that.

\(24x - 1 + 85 = 180\)

Simplify:

\(24x + 84 = 180\)

Subtract 84:

\(24x = 96\)

Divide by 24:

\(x = 4\). Ah, that makes sense. So probably, the figure is a parallelogram, so consecutive angles are supplementary. So angle \(T\) and angle \(U\) are consecutive angles, so they add up to \(180^\circ\).

So step1: Recognize that in a parallelogram, consecutive angles are supplementary. So \(m\angle T + m\angle U = 180^\circ\).

Step2: Substitute the given angles: \( (24x - 1) + 85 = 180 \).

Step3: Simplify: \(24x + 84 = 180\).

Step4: Subtract 84 from both sides: \(24x = 96\).

Step5: Divide by 24: \(x = 4\).

Answer:

\(x = 4\)