NCERT MCQ Solutions for Class 9 Maths Chapter 8 Quadrilaterals

A quadrilateral is a closed figure formed by four line segments. It has four sides, four angles and four vertices. This is the most basic definition of a quadrilateral as given before discussing its properties and types.

A parallelogram is defined as a quadrilateral in which both pairs of opposite sides are equal and parallel. This fundamental property is used in several theorems.

A diagonal of a parallelogram divides it into two congruent triangles. This is proved using the ASA congruence rule and properties of alternate interior angles formed by the diagonal acting as a transversal.

In a parallelogram, opposite sides are equal. This is demonstrated by using the congruence of triangles formed by a diagonal and properties of parallel lines intersected by a transversal.

If both pairs of opposite sides of a quadrilateral are equal, then it is a parallelogram. This is proved by showing triangle congruence using SSS and angle relationships.

The bisectors of the angles of a parallelogram form a rectangle. This is because the angle bisectors intersect to create a figure with all angles equal to 90°.

If the diagonals of a quadrilateral are equal, bisect each other and meet at right angles, then the quadrilateral is a square. This is because these properties uniquely define a square.

In a parallelogram, each pair of opposite angles is equal. This property is derived using triangle congruence and angle properties related to parallel lines cut by a transversal.

The diagonals of a parallelogram bisect each other. This means each diagonal is divided into two equal parts at the point where they intersect. This is a key property in identifying and proving quadrilaterals as parallelograms.

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. This is a useful test to check whether a given quadrilateral is a parallelogram.

Diagonals of a rectangle are equal and bisect each other. This follows from the definition of a rectangle as a parallelogram with equal diagonals. This is proved and stated under special types of parallelograms.

In a rhombus, the diagonals are not only bisecting each other but also intersect at right angles. This is proved using triangle congruence and properties of parallelograms.

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. This follows from the Midpoint Theorem.

The line segment joining the midpoints of the opposite sides of a quadrilateral bisect each other. This property holds true for any quadrilateral.

A rectangle is defined as a parallelogram with each angle equal to 90°. This is proved using the properties of parallel lines and interior angles on the same side of a transversal in the rectangle.

A square has diagonals that are equal in length, bisect each other and intersect at right angles. These properties are a combination of those of a rectangle and a rhombus, both of which are special parallelograms.

A trapezium is defined as a quadrilateral with exactly one pair of opposite sides parallel. This differentiates it from parallelograms which have both pairs of opposite sides parallel.

In a parallelogram opposite sides are equal. So, AB = DC. This equality is a key property of parallelograms and used in several congruence proofs.

In a parallelogram, adjacent angles are supplementary. So, if one angle is 60°, the adjacent angle is 180° – 60° = 120°. This follows from the angle sum property and definition of a parallelogram.

The quadrilateral formed by joining the midpoints of the sides of a quadrilateral PQRS, taken in order, is a rectangle if the diagonals of PQRS are perpendicular. This is because perpendicular diagonals ensure that the resulting figure has all angles equal to 90°.

The quadrilateral formed by joining the midpoints of the sides of a quadrilateral PQRS, taken in order, is a rhombus if the diagonals of PQRS are equal. This is because equal diagonals ensure that the resulting figure has all sides equal in length.

A square is a special parallelogram where diagonals are equal in length and intersect at 90°.

The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. This is illustrated through multiple constructions and activities. 

The converse of the midpoint theorem states that a line drawn through the midpoint of one side and parallel to another side bisects the third side. This is shown through multiple triangle-based proofs.

When a line is drawn from the midpoint of one non-parallel side of a trapezium parallel to the base, it bisects the opposite non-parallel side. 

In any parallelogram, the diagonals bisect each other. This means each diagonal cuts the other into two equal parts at the point of intersection. This is a key test for identifying a parallelogram.

If APB and CQD are two parallel lines, then the bisectors of the angles ∠APQ, ∠BPQ, ∠CQP and ∠PQD form a rectangle. This is because the angle bisectors create right angles at their intersections due to the properties of parallel lines and transversals.

The figure obtained by joining the midpoints of the sides of a rhombus, taken in order, is a rectangle. This is because the diagonals of a rhombus are perpendicular, resulting in a quadrilateral with all angles equal to 90°.

A square has all sides equal like a rhombus and all angles equal like a rectangle. It also has diagonals that are equal in length and intersect at 90°, combining all special properties of parallelograms.

In a rectangle, diagonals bisect each other and are of equal length. This is used in proving triangle congruence and symmetry properties of rectangles as a special type of parallelogram.

If any one angle in a parallelogram is a right angle, then all angles are right angles. Therefore, the parallelogram becomes a rectangle. This result is shown using angle properties of transversals and parallel lines.

The angle bisectors of a parallelogram form a rectangle. This is proved using angle calculations and properties of bisectors and vertically opposite angles formed by intersecting lines.

By joining the midpoints of the three sides of a triangle, four smaller triangles of equal area and congruent shape are formed.

The statement “Opposite angles are bisected by the diagonals” is not true for a general parallelogram. While the diagonals bisect each other, they do not necessarily bisect the opposite angles unless the parallelogram is a rhombus.

In a triangular prism, the lateral faces are rectangles. This is because the corresponding sides of the two parallel triangular bases are connected by rectangular faces.