Question

complete this sentence with the best of the choices given. if the diagonals ______, then a quadrilateral is a parallelogram. after you pick your answer press go. a. are parallel b. are perpendicular c. are congruent d. bisect each other e. intersect

Explanation:

• Option A: Diagonals cannot be parallel. Diagonals are line - segments connecting non - adjacent vertices of a quadrilateral. Parallel lines do not intersect, but diagonals of a quadrilateral (except in some degenerate cases) intersect.
• Option B: Diagonals being perpendicular is a property of some special quadrilaterals like rhombus (a type of parallelogram) and square (a type of parallelogram), but it is not a general condition for a quadrilateral to be a parallelogram. For example, a kite has perpendicular diagonals but is not a parallelogram.
• Option C: Diagonals being congruent is a property of rectangles (a type of parallelogram) and isosceles trapezoids (not a parallelogram). So, congruent diagonals do not guarantee a quadrilateral is a parallelogram.
• Option D: One of the fundamental theorems in geometry states that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Let the quadrilateral be \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at point \(O\). If \(AO = OC\) and \(BO=OD\), then \(\triangle AOB\cong\triangle COD\) (by the Side - Angle - Side (SAS) congruence criterion, since \(\angle AOB=\angle COD\) (vertically opposite angles) and \(AO = OC\), \(BO = OD\)). Then \(\angle OAB=\angle OCD\) (corresponding parts of congruent triangles), so \(AB\parallel CD\). Similarly, \(\triangle AOD\cong\triangle BOC\), and \(AD\parallel BC\).
• Option E: Diagonals of any non - degenerate quadrilateral (except some special cases in 3D) intersect. For example, a trapezoid (which is not a parallelogram in general) has intersecting diagonals.

Answer:

D. bisect each other