Fill in the blanks regarding quadrilaterals and answer questions about a rectangle's angles and sides.

1. Fill in the blanks.

• i. A quadrilateral in which all sides and all angles are equal is a square.
• ii. In a rhombus, opposite angles are equal and all sides are equal.
• iii. A quadrilateral in which two pairs of adjacent sides are equal, but the opposite sides are unequal, is called a kite.
• iv. A quadrilateral whose one pair of opposite sides is parallel and the other pair of opposite sides is non-parallel is called a trapezium (or trapezoid).
• v. The diagonals of a rhombus bisect each other at right angles.

2. In Figure 1, ABCD is a rectangle. Measure its angles and sides and answer the following questions.

• i. What are the measures of $\angle A, \angle B, \angle C$ and $\angle D$? In a rectangle, all interior angles are right angles.

  $$\angle A = \angle B = \angle C = \angle D = \boxed{90^\circ}$$

• ii. Two sets of line segments are given. Match equal line segments. In a rectangle:

  • Opposite sides are equal.

  • Diagonals are equal and bisect each other.

  • Therefore, the segments from the center to each vertex are equal.

  • a. $\overline{AB}$ matches 2. $\overline{CD}$ (opposite sides)

  • b. $\overline{BC}$ matches 4. $\overline{AD}$ (opposite sides)

  • c. $\overline{AC}$ matches 1. $\overline{BD}$ (diagonals)

  • d. $\overline{OA}$ matches 3. $\overline{OC}$ (halves of a diagonal)

3. Name each of the following parallelograms.

• i. All sides are equal and the diagonals are unequal. This describes a rhombus.

• ii. The adjacent sides are unequal and the diagonals are equal. This describes a rectangle.

• iii. The adjacent sides are equal and the diagonals are equal. This describes a square.

4. In Figure 2, kite ABCD has diagonals AC and BD intersecting at O.

• i. Name three pairs of equal line segments. In a kite:

  • Two pairs of adjacent sides are equal.
  • One diagonal is bisected by the other. The three pairs of equal line segments are:
  • $\boxed{\overline{AB} = \overline{AD}}$
  • $\boxed{\overline{CB} = \overline{CD}}$
  • $\boxed{\overline{BO} = \overline{OD}}$

• ii. What is the measure of $\angle AOB$? In a kite, the diagonals are perpendicular to each other.

  $$\angle AOB = \boxed{90^\circ}$$

• iii. Is $\overline{OB} = \overline{OD}$? Yes, one diagonal of a kite (the one connecting the vertices between the unequal sides) is bisected by the other diagonal. In Figure 2, $\overline{BD}$ is bisected by $\overline{AC}$. The answer is $\boxed{\text{Yes}}$.