Question

find the measure of ∠q in parallelogram qrst.
t
q
5x + 12
3x + 46
s
r
label optional

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)), and opposite angles are equal. Here, \(\angle Q\) and \(\angle S\) are consecutive angles? Wait, no—wait, in parallelogram QRST, sides \(QT\) and \(SR\) are parallel, \(QS\) and \(RT\)? Wait, no, the vertices are Q, R, S, T. So \(QR \parallel ST\) and \(QT \parallel RS\). So \(\angle Q\) and \(\angle S\) are same - side interior angles? Wait, no, actually, in a parallelogram, consecutive angles (adjacent angles) are supplementary. Wait, looking at the diagram, \(\angle Q\) is \(5x + 12\) and \(\angle S\) is \(3x+46\). Wait, no—wait, in parallelogram QRST, \(QT\) and \(SR\) are parallel, and \(QS\) is a transversal? Wait, no, maybe \(\angle Q\) and \(\angle S\) are supplementary? Wait, no, actually, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait, let's check the sides: \(QR\) and \(ST\) are one pair of opposite sides, \(QT\) and \(RS\) are the other pair. So \(\angle Q\) and \(\angle S\) are consecutive angles? Wait, no, \(\angle Q\) and \(\angle R\) are consecutive, \(\angle R\) and \(\angle S\) are consecutive, \(\angle S\) and \(\angle T\) are consecutive, \(\angle T\) and \(\angle Q\) are consecutive. Wait, maybe I made a mistake. Wait, the angles given: \(\angle Q = 5x + 12\) and \(\angle S=3x + 46\). Wait, in a parallelogram, opposite angles are equal. Wait, is \(\angle Q\) opposite to \(\angle S\)? Wait, no, in parallelogram QRST, the vertices are in order Q, R, S, T. So \(\angle Q\) is opposite to \(\angle S\), and \(\angle R\) is opposite to \(\angle T\). Wait, that would mean \(\angle Q=\angle S\). Wait, but that would be if they are opposite. Wait, let's confirm: in a parallelogram, opposite angles are equal. So if \(Q\) and \(S\) are opposite vertices, then \(\angle Q=\angle S\). So set \(5x + 12=3x + 46\).

Step2: Solve for \(x\)

\(5x+12 = 3x + 46\)
Subtract \(3x\) from both sides: \(5x-3x+12=3x - 3x+46\)
\(2x+12 = 46\)
Subtract 12 from both sides: \(2x+12 - 12=46 - 12\)
\(2x=34\)
Divide both sides by 2: \(x = 17\)

Step3: Find the measure of \(\angle Q\)

Substitute \(x = 17\) into the expression for \(\angle Q\): \(5x+12=5\times17 + 12\)
\(5\times17=85\), so \(85 + 12=97\)

Wait, but wait—maybe I was wrong about the angle relationship. Wait, maybe \(\angle Q\) and \(\angle S\) are consecutive angles, so they should be supplementary. Let's check that. If \(\angle Q\) and \(\angle S\) are consecutive, then \((5x + 12)+(3x + 46)=180\)
\(8x+58 = 180\)
\(8x=180 - 58=122\)
\(x=\frac{122}{8}=15.25\), which would give a non - integer, and the first approach gave \(x = 17\) and \(\angle Q = 97\). Wait, let's check the diagram again. The parallelogram is drawn with \(T\) at the top - left, \(S\) at the bottom - left, \(R\) at the bottom - right, \(Q\) at the top - right. So \(QT\) is the right - hand side, \(SR\) is the left - hand side. So \(QT\) is parallel to \(SR\), and \(QS\) is the top - bottom side? Wait, no, the sides: \(QT\) and \(RS\) are parallel, \(QR\) and \(ST\) are parallel. So \(\angle Q\) and \(\angle S\) are same - side interior angles? Wait, no, \(\angle Q\) and \(\angle R\) are consecutive, \(\angle R\) and \(\angle S\) are consecutive, \(\angle S\) and \(\angle T\) are consecutive, \(\angle T\) and \(\angle Q\) are consecutive. Wait, maybe the angles given: \(\angle Q\) is at vertex \(Q\), \(\angle S\) is at vertex \(S\). So in the parallelogram, \(Q\) and \(S\) are opposite vertices, so \(\angle Q=\angle S\). So the first approach is correct. So \(\angle Q = 97^\circ\)

Answer:

\(97^\circ\)