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Problem 194

Prove that the diagonals of a parallelogram bisect each other

Expert verified

Using the properties of a parallelogram and congruent triangles, we prove that triangles AEB and CED, formed by the diagonals of parallelogram ABCD, are congruent using the Side-Angle-Side (SAS) congruence postulate. Since triangle AEB is congruent to triangle CED, we have AE = EC and BE = DE, proving that diagonals AC and BD bisect each other at point E.

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Chapter 12

In the figure, \(\mathrm{ABCD}\) is a parallelogram with diagonals \(\underline{\mathrm{AC}}\) and BD. \(\angle \mathrm{ABC}\) is an obtuse angle. Prove that \(\mathrm{AC}>\mathrm{BD}\).

Chapter 12

Starting with any triangle \(\mathrm{ABC}\); construct the exterior squares \(\mathrm{BCDE}, \mathrm{ACFG}\) and BAHK; then construct parallelograms \(\mathrm{FCDQ}\) and EBKP. Prove \(\triangle \mathrm{PAQ}\) is an isosceles right triangle. (Hint: Draw diagonals \(\mathrm{PB}\) and CQ.)

Chapter 12

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,

Chapter 12

Prove that the bisectors of the angles of a rectangle enclose a square

Chapter 12

If both pairs of opposite angles of a quadrilateral \(\mathrm{ABCD}\) are congruent, then prove that the quadrilateral is a parallelogram. (See Figure.)

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