Question

find the measure of angle c. (just put the number) find ( mangle c ) your answer find qr. (just put the number) find ( qr )

Explanation:

Step1: Solve for \(x\) in the parallelogram \(DEBC\)

In a parallelogram, consecutive angles are supplementary. So, \(m\angle D+m\angle E = 180^{\circ}\).
Given \(m\angle D = 53x\) and \(m\angle E=36x + 2\), we have the equation \(53x+36x + 2=180\).
Combine like - terms: \(89x+2 = 180\).
Subtract 2 from both sides: \(89x=180 - 2=178\).
Divide both sides by 89: \(x=\frac{178}{89}=2\).
Since \(m\angle C=m\angle E\) (opposite angles of a parallelogram are equal), \(m\angle C=36x + 2\).
Substitute \(x = 2\) into \(36x+2\): \(36\times2+2=72 + 2=74\).

Step2: Solve for \(x\) in the parallelogram \(SPQR\)

In a parallelogram, opposite sides are equal. So, \(SP = QR\).
Given \(SP=3x + 8\) and \(QR=-2 + 5x\), we have the equation \(3x+8=-2 + 5x\).
Subtract \(3x\) from both sides: \(8=-2+5x - 3x\).
Simplify: \(8=-2 + 2x\).
Add 2 to both sides: \(2x=8 + 2=10\).
Divide both sides by 2: \(x = 5\).
Substitute \(x = 5\) into \(QR=-2+5x\): \(QR=-2+5\times5=-2 + 25=23\).

Answer:

\(m\angle C = 74\)
\(QR = 23\)