91Ó°ÊÓ

If the diagonals of a parallelogram meet at right angles, prove that the parallelogram is a rhombus.

Use an indirect method of proof to prove: If a diagonal of a parallelogram does not bisect the angles through whose vertices the diagonal is drawn, the parallelogram is not a rhombus.

Prove that the lines joining the midpoints of a rectangle form a rhombus.

Let \(\mathrm{ABCD}\) be a rhombus and let its diagonals, \(\underline{\mathrm{AC}}\) and \(\underline{\text { BD }}\), intersect in point \(E\), Prove that \(D E=B E\).

Let \(\mathrm{ABCD}\) be a rhombus. Prove that diagonal \(\underline{\mathrm{AC}}\) bisects \(\angle \mathrm{A}\).

Given: \(\underline{\mathrm{AN}}\) is an angle bisector of \(\triangle \mathrm{ABC} ; \mathrm{CNBH}\); \(\underline{\mathrm{HA}} \perp \underline{\mathrm{AN}} ; \mathrm{CAR} ; \underline{\mathrm{HS}}\|\underline{\mathrm{AB}} ; \mathrm{ABP} ; \underline{\mathrm{HP}}\| \underline{\mathrm{CA}}\). Prove; quadrilateral APHR is a rhombus.

We are given rhombus ABCD with its diagonals drawn. F is the midpoint of \(\underline{D E}, G\) is the midpoint of \(\underline{B E}\) and \(H\) is a point on \(\underline{\mathrm{AE}} .\), Prove that \(\Delta \mathrm{FGH}\) is an isosceles triangle.

In rhombus \(W X Y Z ; A, B\), and \(C\) are the midpoints of \(\underline{W X}\), \(\underline{X Y}\), and \(\underline{Y Z}\), respectively. Proves \(\triangle \mathrm{ABC}\) is a right triangle.

In the accompanying figure, \(\triangle \mathrm{ABC}\) is given to be an isosceles right triangle with \(\angle \mathrm{ABC}\) a right angle and \(\underline{A B} \cong \underline{B C}\). Line segment \(\underline{B D}\), which bisects \(C A\), is extended to \(\mathrm{E}\), so that \(\underline{\mathrm{BD}} \cong \underline{\mathrm{DE}}\). Prove \(\mathrm{BAEC}\) is a sauare,

Given; Square PQRS, with \(\mathrm{N}\) on \(\underline{\mathrm{RS}}\) so that \(\underline{\mathrm{TS}} \cong \underline{\mathrm{SN}}\). Prove: \(\mathrm{m} \angle \mathrm{STN}=3(\mathrm{~m} \angle \mathrm{NTR})\).