Properties of Parallelograms


In the diagram to the right, quadrilateral \(ABCD\) is a parallelogram and \(M\) is the midpoint of \(\overline{DC}.\)

If the area of \(\triangle BNC\) is \(42,\) what is the area of \(\triangle MNC?\)

In the above diagram, \(\angle BAD \) = \(\angle DCB \) and \(\angle ADC \) = \(\angle CBA .\) If the length of \(\overline{AB} \) is \(21\) and the length of \(\overline{AD} \) is \(16,\) what is the length of \(\overline{BC} \)?

In the quadrilateral \(ABCD\) above, \(\overline{AB} \) is parallel to \(\overline{DC}\) and \(\overline{AD} \) is parallel to \(\overline{BC}.\) If the length of \(\overline{DO}\)= \(41,\) then what is the length of \(\overline{BO}?\)

In the above diagram, quadrilateral \(ABCD\) is a parallelogram. If \(\alpha = 69 ^\circ\) and \(\beta = 55 ^\circ ,\) what is \(\angle BAC \)?

In the quadrilateral \(ABCD\) above, let \(\lvert\overline{AB} \rvert\) denote the length of \(\overline{AB}.\) If \(\overline{AB} \) is parallel to \(\overline{DC},\) \(\lvert\overline{AB}\rvert=\lvert\overline{DC}\rvert = 23,\) and \(\lvert\overline{AD}\rvert=15,\) what is \(\lvert\overline{BC}\rvert?\)


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