Question

quadrilateral abcd has the following properties:

• diagonal ac is perpendicular to diagonal bd.
• diagonal ac bisects angle dab.
• opposite sides ab and cd are congruent.

based on this information, which classification of quadrilateral abcd is most justified, and why?
a kite, because the diagonals are perpendicular and one diagonal bisects an angle.
a parallelogram, because opposite sides are congruent and the diagonals are perpendicular.
an isosceles trapezoid, because only one pair of opposite sides is congruent and diagonals are perpe-
a rhombus, because one diagonal is an angle bisector and opposite sides are congruent.

Explanation:

Step1: Analyze option A

A kite has two pairs of adjacent sides equal. The given information about opposite sides \(AB\) and \(CD\) being congruent does not fit the property of a kite.

Step2: Analyze option B

In a parallelogram, diagonals bisect each other but are not necessarily perpendicular. So, this option is incorrect.

Step3: Analyze option C

An isosceles trapezoid has one pair of parallel sides and the non - parallel sides equal. The given information does not match the properties of an isosceles trapezoid.

Step4: Analyze option D

A rhombus has opposite sides congruent (as it is a parallelogram) and the diagonals are perpendicular. Also, a diagonal of a rhombus bisects the angles.

Answer:

D. A rhombus, because one diagonal is an angle bisector and opposite sides are congruent.