Question

5 multiple choice 9 points
which statement is true about every parallelogram?
two pairs of opposite sides are congruent.
the diagonals are perpendicular to each other.
the interior angles are all congruent.
all four sides are congruent.

6 multiple choice 9 points
in the diagram below, parallelogram abcd has diagonals \\(\overline{ac}\\) and \\(\overline{bd}\\) that intersect at point e.
diagram of parallelogram abcd with diagonals intersecting at e
which expression is not always true?
a. \\(\overline{ac} \cong \overline{db}\\) c. \\(\angle dae \cong \angle bce\\)
b. \\(\overline{de} \cong \overline{eb}\\) d. \\(\angle dec \cong \angle bea\\)
options: b, a, c, d

Explanation:

Question 5

Step1: Recall parallelogram properties

A parallelogram has opposite sides congruent and parallel, opposite angles congruent, diagonals bisect each other.

Step2: Analyze each option

• Option 1: Two pairs of opposite sides are congruent. This is a fundamental property of parallelograms.
• Option 2: Diagonals perpendicular? Only in rhombus (a type of parallelogram), not all parallelograms.
• Option 3: Interior angles all congruent? Only in rectangles (a type of parallelogram), not all.
• Option 4: All four sides congruent? Only in rhombus, not all parallelograms.

Step1: Recall parallelogram diagonal properties

In a parallelogram, diagonals bisect each other (so \( DE \cong EB \)), opposite sides are parallel (so \( \angle DAE \cong \angle BCE \) by alternate interior angles), vertical angles are congruent (so \( \angle DEC \cong \angle BEA \)). Diagonals are not necessarily congruent (that's for rectangles), so \( \overline{AC} \cong \overline{DB} \) is not always true.

Step2: Match with options

Option a (\( \overline{AC} \cong \overline{DB} \)) is not always true.

Answer:

Two pairs of opposite sides are congruent. (The first option)

Question 6