Question

multiple choice 1 point
given three distinct quadrilaterals, a square, a rectangle, and a rhombus, which quadrilaterals must have perpendicular diagonals?
the rectangle and the square
the rectangle, the rhombus, and the square
the rhombus, only
the rhombus and the square

multiple choice 1 point
abcd is a parallelogram (not a rhombus or rectangle). which of the following statements regarding this parallelogram is possibly false?
\\( \overline { a c } \perp \overline { b d } \\)
\\( \overline { b c } \cong \overline { c d } \\)
\\( \angle b \cong \angle d \\)
\\( \overline { b a } \parallel \overline { d c } \\)

Explanation:

Step1: Recall properties of quadrilaterals

• Rectangle: Diagonals are equal in length but not necessarily perpendicular.
• Rhombus: Diagonals are perpendicular bisectors of each other.
• Square: A special case of both rectangle and rhombus, so diagonals are perpendicular.

Step2: Analyze each option

• Option 1: Rectangle's diagonals are not perpendicular. So this option is wrong.
• Option 2: Rectangle's diagonals are not perpendicular. So this option is wrong.
• Option 3: Square (a special rhombus) also has perpendicular diagonals. So this option is wrong.
• Option 4: Rhombus has perpendicular diagonals and square (a special rhombus) also has. This option is correct.

Step1: Recall properties of parallelograms

• In a parallelogram \(ABCD\), opposite sides are parallel (\(\overline{BA}\parallel\overline{DC}\)), opposite angles are equal (\(\angle B\cong\angle D\)), and adjacent sides are not necessarily congruent (\(\overline{BC}\cong\overline{CD}\) is only true for rhombus).
• Diagonals are perpendicular (\(\overline{AC}\perp\overline{BD}\)) only for rhombus (and square which is a special case).

Step2: Analyze each option

• \(\overline{AC}\perp\overline{BD}\): True for rhombus (and square), but since \(ABCD\) is not a rhombus, this is possibly false.
• \(\overline{BC}\cong\overline{CD}\): True for rhombus, but since \(ABCD\) is not a rhombus, this is possibly false.
• \(\angle B\cong\angle D\): True for all parallelograms.
• \(\overline{BA}\parallel\overline{DC}\): True for all parallelograms.

Answer:

the rhombus and the square