11. Two lines AB and CD are parallel to each other. The transversal BD

1.	11. Two lines AB and CD are parallel to each other. The transversal BD bisects the other transversalAC. Prove that AC also bisects BD
Answer» Let the intersection of BD and AC be point E. Angles BAC and DCA are congruent [alternate interior angles]Angles BAC and BAE are congruent [same angle]Angles BAE and DCA are congruent [transitivity of congruence]Angles DCA and DCE are congruent [same angle]Angles BAE and DCE are congruent [transitivity of congruence]Angles AEB and CED are congruent [vertical angles]AE = CE [definition of bisection]triangles AEB and CED are congruent [angle, side, angle]BE = DE [corresponding sides of congruent triangles]AC bisects BD [definition of bisection]

11. Two lines AB and CD are parallel to each other. The transversal BD bisects the other transversalAC. Prove that AC also bisects BD

Answer»

Let the intersection of BD and AC be point E.

Angles BAC and DCA are congruent [alternate interior angles]Angles BAC and BAE are congruent [same angle]Angles BAE and DCA are congruent [transitivity of congruence]Angles DCA and DCE are congruent [same angle]Angles BAE and DCE are congruent [transitivity of congruence]Angles AEB and CED are congruent [vertical angles]AE = CE [definition of bisection]triangles AEB and CED are congruent [angle, side, angle]BE = DE [corresponding sides of congruent triangles]AC bisects BD [definition of bisection]

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