Given that $ABCD$ is a parallelogram, where $AB$ is parallel to $CD$ and $AD$ is p

Question

Given that $ABCD$ is a parallelogram, where $AB$ is parallel to $CD$ and $AD$ is parallel to $BC$, find the value(s) of the unknown(s) in each of the following figures, stating your reasons clearly. Source consistency note: In part (i), the exact angle labels are inconsistent. The angle chase gives $\angle BAO=59^\circ,\angle ABO=39^\circ,\angle BCO=40^\circ$. Thus $AO/BO=\sin39^\circ/\sin59^\circ$, whereas $CO/BO=\sin42^\circ/\sin40^\circ$. These are unequal, contradicting the diagonal-bisection property $AO=CO$. Part (ii) is unaffected. The numerical answers below are conditional angle-chasing results, not angles of a figure satisfying every label.

Accepted Answer

(i) $x=39,y=40,z=59$; (ii) $x=44,y=92,z=136$. The part (i) results are conditional; the exact source labels conflict.

Quadrilaterals

Quadrilaterals

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