Question

it is given that quadrilateral abcd is a parallelogram. which of the following statements must be true? (1 point)
○ $overline{ae}$ is congruent to $overline{eb}$.
○ $overline{ae}$ is congruent to $overline{ce}$.
○ $overline{de}$ is congruent to $overline{ec}$.
○ $overline{ca}$ is congruent to $overline{db}$.

Explanation:

Step1: Recall parallelogram property

In a parallelogram, the diagonals bisect each other. In parallelogram \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at \(E\), we have \(AE = EC\) and \(DE=EB\).

Step2: Analyze each option

• Option 1: \(\overline{AE}\) is congruent to \(\overline{EB}\) is not always true for a general parallelogram.
• Option 2: Since the diagonals of a parallelogram bisect each other, \(\overline{AE}\) is congruent to \(\overline{CE}\). This is a property of parallelograms.
• Option 3: \(\overline{DE}\) is congruent to \(\overline{EC}\) is not always true for a general parallelogram.
• Option 4: \(\overline{CA}\) is congruent to \(\overline{DB}\) is not always true. A parallelogram's diagonals are equal in length only when it is a rectangle.

Answer:

\(\overline{AE}\) is congruent to \(\overline{CE}\)