This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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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 A, B, C and D? In a rectangle, all interior angles are right angles. A = B = C = D = 90^ 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. AB matches 2. CD (opposite sides) b. BC matches 4. AD (opposite sides) c. AC matches 1. BD (diagonals) d. OA matches 3. 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: AB = AD CB = CD BO = OD ii. What is the measure of AOB? In a kite, the diagonals are perpendicular to each other. AOB = 90^ iii. Is OB = 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, BD is bisected by AC. The answer is Yes.