Question

Parallelogram ABCD has diagonals AC and BD that intersect at point E as shown. Which of the following statements is NOT always true?

a) AC is equal to DB.
b) ∠DAE is congruent to ∠BCE.
c) DE is congruent to EB.
d) ∠DEC is congruent to ∠BEA.

Answer 1

In a parallelogram, the diagonals bisect each other, meaning that they divide each other into two equal parts. Statement a), b), and c) are always true, but statement d) is not always true.

Step-by-step explanation:

In a parallelogram, the diagonals bisect each other, meaning that they divide each other into two equal parts. This property holds true for parallelogram ABCD as well. So, statement a) AC is equal to DB, b) ∠DAE is congruent to ∠BCE, and c) DE is congruent to EB are always true. However, statement d) ∠DEC is congruent to ∠BEA is not always true.

To understand why statement d) is not always true, let's consider a specific example. Suppose in parallelogram ABCD, angle C is greater than angle D. In this case, the sum of angles ∠DEC and ∠BEA will be less than 180 degrees, making them unequal. Therefore, statement d) is not always true.