QUADRILATERALS
Frequently Asked Questions
A quadrilateral is a closed figure with four sides, four angles, and four vertices. The sum of all interior angles of a quadrilateral is \(360^\circ\).
Parallelogram, Rectangle, Rhombus, Square, Trapezium, and Kite.
Opposite sides are equal and parallel, opposite angles are equal, and diagonals bisect each other.
The sum of all interior angles of a quadrilateral \(= 360^\circ\).
A quadrilateral whose opposite sides are parallel is called a parallelogram.
A rectangle is a parallelogram with all angles equal to \(90^\circ\).
Opposite sides are equal and parallel, all angles are \(90^\circ\), and diagonals are equal and bisect each other.
All sides are equal, opposite angles are equal, diagonals bisect each other at right angles.
All sides are equal, all angles are \(90^\circ\), diagonals are equal, and bisect each other at right angles.
A quadrilateral with one pair of opposite sides parallel is called a trapezium.
A trapezium in which the non-parallel sides are equal in length.
A quadrilateral with two pairs of adjacent sides equal and diagonals intersecting at right angles.
The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of its length.
A line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side.
(i) Opposite sides are equal, <br>(ii) Opposite sides are parallel,<br> (iii) Diagonals bisect each other, <br>(iv) One pair of opposite sides is equal and parallel.
Diagonals bisect each other but are not necessarily equal.
They are equal and bisect each other.
They bisect each other at right angles (perpendicular).
They are equal in length and perpendicular to each other.
Sum of interior angles = \((n - 2) \times \(180^\circ\)\), where\ (n\) = number of sides.
Two sides sharing a common vertex are adjacent sides. The sides that do not share a vertex are opposite sides.
Angles sharing a common arm are adjacent, and those that do not share an arm are opposite.
A line segment joining any two non-adjacent vertices of a quadrilateral.
Two diagonals.
A quadrilateral with all interior angles less than \(180^\circ\).
A quadrilateral in which one interior angle is more than \(180^\circ\).
A quadrilateral whose all vertices lie on a single circle.
The sum of the opposite angles is \(180^\circ\).
A rhombus has all sides equal, while a kite has two pairs of adjacent equal sides.
Diagonals are perpendicular; one diagonal bisects the other.
By showing any one of the conditions: opposite sides/angles are equal or diagonals bisect each other.
\(\angle A + \angle B + \angle C + \angle D = \(360^\circ\)\.)
Opposite sides are both equal and parallel.
By showing triangles formed by diagonals are congruent using SSS or ASA rule.
Used in coordinate geometry and triangle problems to find unknown lengths or prove parallelism.
Construction (walls, tiles), architecture, computer graphics, road design, and frames.
A rectangle has equal diagonals and all angles = \(90^\circ\), while a parallelogram doesn’t necessarily have these properties.
In a rhombus all sides are equal, but in a parallelogram opposite sides are equal.
A square is a special parallelogram with all sides and angles equal.
They are used to show the bisecting property and congruence of opposite triangles.
(i) Sum of angles = \(360^\circ\); (ii) Area formulas vary: for parallelogram \(A = b \times h\); for rectangle \(A = l \times b\); for square \(A = a^2\).
By showing the two smaller triangles are congruent using ASA or SAS rule.
Reflection, rotation, and translation to prove congruency and parallelism.
SSS, SAS, ASA, and AAS rules.
(i) Diagonals of a parallelogram bisect each other. <br>(ii) Converse of above theorem. <br>(iii) Mid-point theorem and its converse.
If one angle = \(90^\circ\), the parallelogram is a rectangle.
If all sides are equal, it’s a rhombus.
If all sides are equal and all angles = \(90^\circ\), it’s a square.
Mid-point theorem, properties of parallelogram, rectangle, rhombus, and proving equal angles or sides using congruence.
Proof-based theorems, fill-in-the-blanks on properties, and short questions on identifying shapes.
Four major theorems: Diagonals bisect, Converse of bisect theorem, Mid-Point Theorem, and its Converse.
Mid-Point Theorem and its converse.
Group them by type: parallelogram \(\Rightarrow\) rectangle \(\Rightarrow\) rhombus \(\Rightarrow\) square (progressively adding equal angles and sides).
Table tops, window frames, slanted walls, and tilted ladders.
Used in engineering and architecture to create stable and symmetrical designs.
Trapezium has only one pair of parallel sides; parallelogram has two pairs.
\(A = base \times height\).
\(P = 2(l + b)\).
\(A = a^2\).
By paper folding or coordinate geometry plotting.
Geometry tools like compass, protractor, ruler, and coordinate grid.
Geometric operations like rotation, reflection, or translation applied to quadrilaterals.
Clock faces and circular tables.
\(A = \frac{1}{2} (a + b)h\).
Because opposite sides are equal and parallel, forming congruent triangles.
Because all sides are equal, and congruent triangles around diagonals are symmetrical.
Parallelism, congruence, and equality of opposite sides and angles.
Yes, if its adjacent sides are equal.
Yes, if all angles are \(90^\circ\).
Yes, but every rhombus is not a square.
Interior angles are inside the figure, exterior are formed by extending sides outward.
A line that cuts two or more parallel lines.
Alternate interior angles, corresponding angles, and vertically opposite angles.
Solving proof-based and construction geometry problems in CBSE exams.
All properties, theorems, and standard proofs from NCERT Chapter 8 Quadrilaterals.
