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Question

A parallelogram has vertices upper A, upper B, upper C, and upper D. Opposite vertices are connected via diagonal lines that intersect at an interior point labeled upper E.

It is given that quadrilateral ABCD
is a parallelogram. Which of the following statements must be true?

(1 point)
Responses

CA¯¯¯¯¯¯¯¯
is congruent to DB¯¯¯¯¯¯¯¯
.
line segment cap c cap A is congruent to line segment cap d cap b. - no response given

AE¯¯¯¯¯¯¯¯
is congruent to EB¯¯¯¯¯¯¯¯
.
line segment cap A cap e is congruent to line segment cap e cap b. - no response given

AE¯¯¯¯¯¯¯¯
is congruent to CE¯¯¯¯¯¯¯¯
.
line segment cap A cap e is congruent to line segment cap c cap e. - no response given

DE¯¯¯¯¯¯¯¯
is congruent to EC¯¯¯¯¯¯¯¯
.
line segment cap d cap e is congruent to line segment cap e cap c. - incorrect
Incorrect

These two segments do not form a diagonal.
Remaining Attempts : 1
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Answer 1

In a parallelogram, the following properties hold true:

1. Opposite sides are congruent: \( \overline{AB} \cong \overline{CD} \) and \( \overline{AD} \cong \overline{BC} \).
2. The diagonals bisect each other.

Given that quadrilateral ABCD is a parallelogram, we can evaluate the statements regarding segments and points related to the diagonals.

1. Since diagonals \(AC\) and \(BD\) bisect each other at point E, we can confirm the following:

   • \( \overline{AE} \cong \overline{EC} \) (the diagonals are bisected)
   • \( \overline{BE} \cong \overline{ED} \) (the diagonals are bisected)

2. For the options presented:

   • CA is congruent to DB: This statement is not true; rather, \( \overline{AB} \cong \overline{CD} \) and \( \overline{AD} \cong \overline{BC} \).
   • AE is congruent to EB: This statement is not true; it should be \( \overline{AE} \cong \overline{EC} \).
   • AE is congruent to CE: This statement is true, as explained above.
   • DE is congruent to EC: This statement is incorrect; it is actually \( \overline{DE} \cong \overline{BE} \).

Based on the analysis, the statement that must be true is:

AE is congruent to CE: \( \overline{AE} \cong \overline{CE} \).