Question

consider the diagram of quadrilateral abcd which is not drawn to scale.
which two statements must be true based on the information indicated by the diagram?
choose 2 answers:
a $angle cdb cong angle bcd$
b $angle adb cong angle cdb$
c $overline{ac} cong overline{bd}$
d $overline{ae} cong overline{ce}$

Explanation:

1. For option d: In quadrilateral \(ABCD\) (a parallelogram, as diagonals bisect each other or sides are parallel), \(AB \parallel CD\) and \(AD \parallel BC\). \(\angle ABE\) and \(\angle CDE\) are alternate - interior angles formed by the transversal \(BD\) cutting the parallel lines \(AB\) and \(CD\). So, by the alternate - interior angles theorem, \(\angle ABE\cong\angle CDE\).
2. For option b: In a parallelogram, adjacent angles are supplementary, but \(\angle CDE\) and \(\angle BCD\) are not necessarily congruent. In a rectangle (a special parallelogram), \(\angle BCD = 90^{\circ}\), but \(\angle CDE\) is not necessarily \(90^{\circ}\) unless it's a rectangle with specific properties.
3. For option c: In a general parallelogram, the diagonals are not congruent. Diagonals are congruent only in special parallelograms like rectangles.
4. For option d: In a parallelogram, the diagonals bisect each other. So, \(AE = CE\) because the diagonals \(AC\) and \(BD\) intersect at \(E\) and bisect each other.

Answer:

d. \(\angle ABE\cong\angle CDE\)
e. \(AE = CE\)