Question

1. solve for x.

(3x + 5)°
(9x - 17)°

Explanation:

Step1: Identify the property

In a parallelogram, consecutive angles are supplementary, and also, in this case, angle \( Q \) and angle \( T \) seem to be related (probably consecutive angles or maybe alternate? Wait, looking at the diagram, it's a parallelogram \( QRST \) (assuming), so consecutive angles are supplementary? Wait, no, actually, in a parallelogram, consecutive angles are supplementary, but also, if it's a parallelogram, angle \( Q \) and angle \( T \) – wait, maybe it's a parallelogram, so angle \( Q \) and angle \( T \) are same - side interior angles? Wait, no, the diagram shows a parallelogram, so adjacent angles are supplementary? Wait, no, actually, in a parallelogram, consecutive angles are supplementary, but also, if we look at the angles, \( \angle Q=(3x + 5)^\circ \) and \( \angle T=(9x-17)^\circ \). Wait, maybe in a parallelogram, consecutive angles are supplementary? Wait, no, actually, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait, but maybe here, \( \angle Q \) and \( \angle T \) are same - side interior angles? Wait, no, the diagram: \( Q, R, T \) – maybe it's a parallelogram, so \( QR \parallel ST \) and \( QT \parallel RS \). So angle \( Q \) and angle \( T \) – wait, maybe they are same - side interior angles? Wait, no, maybe the two angles are supplementary? Wait, no, wait, in a parallelogram, consecutive angles are supplementary. Wait, but maybe the two angles given are same - side interior angles, so they are supplementary? Wait, no, wait, let's re - examine. The figure is a parallelogram, so \( \angle Q \) and \( \angle T \) – wait, maybe \( \angle Q \) and \( \angle T \) are supplementary? Wait, no, in a parallelogram, consecutive angles are supplementary. Wait, maybe the two angles are equal? Wait, no, the markings: the angle at \( T \) has a double arc, maybe indicating that \( \angle Q \) and \( \angle T \) are supplementary? Wait, no, maybe I made a mistake. Wait, let's think again. In a parallelogram, consecutive angles are supplementary. So if \( \angle Q=(3x + 5)^\circ \) and \( \angle T=(9x - 17)^\circ \), and they are consecutive angles, then \( (3x + 5)+(9x-17)=180 \)? Wait, no, that would be if they are supplementary. But maybe they are equal? Wait, no, the double arc at \( T \) – maybe \( \angle Q \) and \( \angle T \) are same - side interior angles, so they are supplementary? Wait, no, maybe the two angles are equal? Wait, no, let's check the problem again. Wait, the diagram is a parallelogram, so opposite angles are equal, and consecutive angles are supplementary. Wait, maybe the angle at \( Q \) and the angle at \( T \) are same - side interior angles, so they are supplementary. Wait, no, let's solve both cases.

Case 1: If they are supplementary: \( (3x + 5)+(9x-17)=180 \)
\( 12x-12 = 180 \)
\( 12x=192 \)
\( x = 16 \)

Case 2: If they are equal: \( 3x + 5=9x-17 \)
\( 5 + 17=9x-3x \)
\( 22 = 6x \)
\( x=\frac{11}{3}\approx3.67 \)

But since it's a parallelogram, and the angle at \( T \) has a double arc, maybe the two angles are supplementary? Wait, no, maybe I was wrong. Wait, let's look at the diagram again. The figure is a parallelogram, so \( QT \) is a side, \( QR \) and \( ST \) are sides. So angle \( Q \) and angle \( T \) – if \( QT \) is a transversal cutting \( QR \) and \( ST \), then angle \( Q \) and angle \( T \) are same - side interior angles, so they are supplementary. Wait, no, same - side interior angles are supplementary when the lines are parallel. Since \( QR \parallel ST \), then \( \angle Q+\angle T = 180^\circ \). Wait, but that gives \( x = 16 \). But let's check the other case. If they are equal, \( x=\frac{11}{3}\), but that seems less likely. Wait, maybe the two angles are equal? Wait, the double arc at \( T \) – maybe it's indicating that \( \angle T \) is equal to another angle, but \( \angle Q \) is given as \( (3x + 5)^\circ \). Wait, maybe I made a mistake in the property. Wait, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So if \( \angle Q \) and \( \angle T \) are consecutive angles, then they are supplementary. So let's go with that.

Step2: Set up the equation

If \( \angle Q \) and \( \angle T \) are supplementary (consecutive angles in a parallelogram), then:

\( (3x + 5)+(9x-17)=180 \)

Combine like terms:

\( 3x+9x + 5-17=180 \)

\( 12x-12 = 180 \)

Add 12 to both sides:

\( 12x=180 + 12=192 \)

Divide both sides by 12:

\( x=\frac{192}{12}=16 \)

Wait, but wait, maybe the two angles are equal? Let's check the other case. If \( 3x + 5=9x-17 \)

Subtract \( 3x \) from both sides:

\( 5 = 6x-17 \)

Add 17 to both sides:

\( 22 = 6x \)

\( x=\frac{22}{6}=\frac{11}{3}\approx3.67 \)

But the diagram has a double arc at \( T \), maybe indicating that \( \angle T \) is equal to \( \angle R \), and \( \angle Q \) is equal to \( \angle S \). Wait, no, maybe the two angles given are same - side interior angles, so they are supplementary. Wait, but let's re - examine the problem. The problem says "Solve for \( x \)" with the diagram of a parallelogram. In a parallelogram, consecutive angles are supplementary. So \( \angle Q \) and \( \angle T \) are consecutive angles, so they should be supplementary. So the first case is correct.

Wait, but maybe I made a mistake. Wait, let's check the angles. If \( x = 16 \), then \( \angle Q=3\times16 + 5=53^\circ \), \( \angle T=9\times16-17=144 - 17 = 127^\circ \), and \( 53+127 = 180 \), which is supplementary. So that works.

If \( x=\frac{11}{3}\), then \( \angle Q=3\times\frac{11}{3}+5 = 16^\circ \), \( \angle T=9\times\frac{11}{3}-17=33 - 17 = 16^\circ \), which are equal. But the diagram has a double arc at \( T \), maybe indicating that \( \angle T \) is equal to another angle, but \( \angle Q \) is not marked the same. So the supplementary case is more likely.

Answer:

\( x = 16 \)