Quadrilaterals, Polygons and Symmetry
Quadrilaterals, Polygons and Symmetry
Quadrilaterals give 2 questions and are examined through their properties rather than through calculation. Most of the marks turn on knowing exactly which property belongs to which figure, since the family shares many of them and questions exploit the overlap.
The Quadrilateral Family
| Figure | Defining properties |
|---|---|
| Trapezium | One pair of opposite sides parallel |
| Parallelogram | Both pairs of opposite sides parallel and equal |
| Rectangle | Parallelogram with all angles 90° |
| Rhombus | Parallelogram with all sides equal |
| Square | Both a rectangle and a rhombus |
| Kite | Two pairs of adjacent sides equal |
The family is nested: every square is a rectangle and a rhombus, every rectangle and rhombus is a parallelogram, and every parallelogram is a trapezium. The reverse is never true, and statement-based questions are built entirely on that one-way relationship.
✗ Every rhombus is a square | ✓ Every square is a rhombus, but a rhombus need not have right angles
Diagonal Properties
| Figure | Diagonals bisect each other | Diagonals equal | Diagonals perpendicular |
|---|---|---|---|
| Parallelogram | Yes | No | No |
| Rectangle | Yes | Yes | No |
| Rhombus | Yes | No | Yes |
| Square | Yes | Yes | Yes |
| Kite | One bisects the other | No | Yes |
This single table answers more questions than any other in the topic. The pattern is worth stating aloud: equal diagonals mean right angles are present; perpendicular diagonals mean equal sides are present, and a square has both because it has both.
A diagonal of a parallelogram divides it into two congruent triangles, so it also halves the area — a fact used constantly in area questions.
Polygons
Interior angle sum = (n − 2) × 180° | Number of diagonals = n(n − 3) ÷ 2
A hexagon has 6(3)/2 = 9 diagonals. For a regular polygon, each interior angle is (n − 2)180/n and each exterior angle is 360/n, and the two must add to 180 degrees, which is a useful self-check.
Symmetry
A figure has line symmetry if a line divides it into two mirror halves, and rotational symmetry if it looks the same after a rotation of less than a full turn. The order of rotational symmetry is the number of such positions in one full turn.
| Figure | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Rhombus | 2 | 2 |
| Equilateral triangle | 3 | 3 |
| Circle | Infinite | Infinite |
| Parallelogram | 0 | 2 |
The parallelogram is the entry that gets asked, because it has no line of symmetry at all yet has rotational symmetry of order 2. Most candidates assume every symmetric-looking figure has mirror lines.
हर वर्ग समचतुर्भुज है, पर हर समचतुर्भुज वर्ग नहीं।
TRE pointer: The diagonal table is the highest-yield half-page in the geometry block; a single question can be answered from it in five seconds. The parallelogram has zero lines of symmetry fact is the classic trap in the symmetry part, and it is exactly the kind of item a class 7 teacher is expected to have right. With −1/3 for a wrong answer across five options, memorising one table beats reasoning through four figures.
60-Second Recap
- The family nests: square is inside rectangle and rhombus, both inside parallelogram.
- Rectangle has equal diagonals; rhombus has perpendicular diagonals; a square has both.
- A diagonal of a parallelogram makes two congruent triangles and halves the area.
- Interior angle sum is (n−2)180°; number of diagonals is n(n−3)/2.
- A parallelogram has no line of symmetry but rotational symmetry of order 2.
- A square has 4 lines of symmetry and rotational order 4.
Frequently Asked Questions
In which quadrilaterals are the diagonals perpendicular?
In a rhombus, a square and a kite. Diagonals are equal in a rectangle and a square. So a square appears in both lists, since it is both a rectangle and a rhombus. Learning the diagonal table as a grid answers most quadrilateral questions instantly.
How many lines of symmetry does a parallelogram have?
None. Despite looking balanced, a general parallelogram cannot be folded onto itself along any line. It does have rotational symmetry of order two, since a half turn maps it onto itself. This gap between the two kinds of symmetry is exactly what the question tests.
How many diagonals does a polygon of n sides have?
n multiplied by n minus three, all divided by two. A hexagon therefore has nine diagonals and an octagon twenty. The formula counts each diagonal once, which is why the division by two is needed and why omitting it is a common slip.
Is every rectangle a parallelogram?
Yes. A rectangle satisfies every parallelogram property and adds right angles. The reverse is false, since a parallelogram need not have right angles. The same one-way logic runs through the whole family, and statement questions are built on candidates reversing it.