Quadrilaterals — Important Questions

6 cm

Step 1: By the midpoint theorem, the line segment joining midpoints of two sides of a triangle is parallel to the third side and half its length. Step 2: D and E are midpoints of AB and AC respectively. Step 3: Therefore, DE is parallel to BC and DE = (1/2) × BC. Step 4: Since BC = 12 cm, DE = (1/2) × 12 = 6 cm.

Parallelogram

Step 1: If both pairs of opposite sides of a quadrilateral are equal, then it is a parallelogram (Theorem 8.3). Step 2: This is one of the defining properties that makes a quadrilateral a parallelogram. Step 3: While rectangles and rhombi are special types of parallelograms, the most general answer is parallelogram. Step 4: A trapezium has only one pair of parallel sides, so it doesn't fit this description.

180°

Step 1: In a parallelogram, adjacent angles are supplementary (they add up to 180°). Step 2: ∠A and ∠B are adjacent angles in parallelogram ABCD. Step 3: This is because AB || DC and AD is a transversal, making ∠A and ∠B interior angles on the same side. Step 4: Therefore, ∠A + ∠B = 180°.

24 cm²

Step 1: The area of a rhombus is (1/2) × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals. Step 2: Given that the diagonals are 6 cm and 8 cm long. Step 3: Area = (1/2) × 6 × 8 = (1/2) × 48 = 24 cm². Step 4: The fact that diagonals intersect at right angles is already accounted for in this formula.