Question

The angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is 60°. Find the angles of the parallelogram.

Answer 1

$$\begin{gathered} \mathrm{Draw}\mathrm{a}\mathrm{parallelogram}\mathrm{ABCD}. \\ \mathrm{Drop}\mathrm{a}\mathrm{perpendicular}\mathrm{from}\mathrm{B}\mathrm{to}\mathrm{the}\mathrm{side}\mathrm{AD},\mathrm{at}\mathrm{the}\mathrm{point}\mathrm{E}. \\ \mathrm{Drop}\mathrm{perpendicular}\mathrm{from}\mathrm{B}\mathrm{to}\mathrm{the}\mathrm{side}\mathrm{CD},\mathrm{at}\mathrm{the}\mathrm{point}\mathrm{F}. \\ \mathrm{In}\mathrm{the}\mathrm{quadrilateral}\mathrm{BEDF}: \\ \angle \mathrm{EBF}=60°,\angle \mathrm{BED}=90° \\ \angle \mathrm{BFD}=90° \\ \angle \mathrm{EDF}=360°-(60°+90°+90°)=120° \\ \mathrm{In}\mathrm{a}\mathrm{parallelogram},\mathrm{opposite}\mathrm{angles}\mathrm{are}\mathrm{congruent}\mathrm{and}\mathrm{adjacent}\mathrm{angles}\mathrm{are}\mathrm{supplementary}. \\ \mathrm{In}\mathrm{the}\mathrm{parallelogram}\mathrm{ABCD}: \\ \angle \mathrm{B}=\angle \mathrm{D}=120° \\ \angle \mathrm{A}=\angle \mathrm{C}=180°-120°=60° \end{gathered}$$