Rectilinear Figure Class 9 ICSE — Chapter 14
Rectilinear Figure Class 9 ICSE is Chapter 14 of the Mathematics textbook. The chapter defines a rectilinear figure and a polygon, then develops the angle-sum formulas that hold for every polygon. It ends with the special quadrilaterals — trapezium, parallelogram, rectangle, rhombus and square — and the theorems that prove their properties. Read each section in the order the book presents it.
Download Rectilinear Figures Class 9 ICSE Mathematics Chapter 14 PDF
Download the Rectilinear Figures Class 9 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 19 pages, 5 sections, 40 questions.
What is inside Chapter 14
19 pages · 5 sections · 13 worked examples · 40 exercise questions
| Section | Title |
|---|---|
| 14.1 | INTRODUCTION |
| 14.2 | NAMES OF POLYGONS |
| 14.3 | REGULAR POLYGON |
| 14.4 | QUADRILATERALS |
| 14.5 | SPECIAL KINDS OF QUADRILATERALS |
- EXERCISE 14(A): 11 questions
- EXERCISE 14(B): 10 questions
- EXERCISE 14(C): 19 questions
14.1 INTRODUCTION
This section sets three definitions that the whole chapter rests on. Rectilinear means along a straight line. A plane figure bounded by straight lines is called a rectilinear figure. A closed plane figure bounded by at least three line segments is called a polygon.
The word “rectilinear” matters because it tells you every side is a straight segment. No curve is allowed in a polygon. Triangles, quadrilaterals and pentagons are all polygons, and all are rectilinear figures.
These definitions explain how the chapter is organised. First you learn how polygons are named and how their angles behave. Then you study quadrilaterals as a special family. For the full list of chapters in this book, visit the Class 9 Maths hub.
14.2 NAMES OF POLYGONS
A polygon is named by the number of sides it has. The book lists the names from 3 to 8 sides.
| Number of sides | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| Name | Triangle | Quadrilateral | Pentagon | Hexagon | Heptagon | Octagon |
The names share Greek prefixes you already know. Tri means 3, quad means 4, pent means 5, hex means 6, hept means 7, oct means 8. Learn the six prefixes in order and the table writes itself.
Polygons split into two types. A convex polygon has every angle less than 180°. A concave polygon has at least one angle greater than 180°. Unless a question says otherwise, “polygon” means a convex polygon.
Two formulas do most of the work in this section. The sum of the interior angles of an n-sided polygon is (2n - 4) right angles, which is (2n - 4) \times 90^\circ. The sum of the exterior angles, formed by producing the sides in order, is always 360^\circ.
Worked pattern: find the number of sides when the sum of interior angles is 1260°.
Step 1: Write the interior angle sum formula.
(2n - 4) \times 90^\circ = 1260^\circ
Step 2: Divide both sides by 90.
2n - 4 = 14
Step 3: Solve for n.
2n = 18 \Rightarrow n = 9
Final answer: the polygon has 9 sides.
A common mistake is confusing the two sums. The interior sum grows with n. The exterior sum stays fixed at 360° for every polygon. Another mistake is forgetting to convert right angles into degrees.
14.3 REGULAR POLYGON
A regular polygon has all sides equal, all interior angles equal and all exterior angles equal. Three formulas follow from that definition.
- Each interior angle = \frac{(2n - 4) \times 90^\circ}{n}
- Each exterior angle = \frac{360^\circ}{n}
- If each exterior angle is x^\circ, the number of sides is \frac{360}{x}
At every vertex, the interior and exterior angles add to 180°. As the number of sides grows, each interior angle grows and each exterior angle shrinks. That is why a regular octagon has a larger interior angle than a regular pentagon.
Worked pattern 1: find the interior angle of a regular polygon with 15 sides.
Step 1: Substitute n = 15 into the interior angle formula.
\frac{(2 \times 15 - 4) \times 90^\circ}{15} = \frac{26 \times 90^\circ}{15} = 156^\circ
Step 2: Check with the exterior angle.
\frac{360^\circ}{15} = 24^\circ, \quad 156^\circ + 24^\circ = 180^\circ
Final answer: each interior angle is 156°.
Worked pattern 2: the exterior angles of two regular polygons with n and (n + 1) sides differ by 4°.
Step 1: Write the difference of the two exterior angles.
\frac{360^\circ}{n} - \frac{360^\circ}{n+1} = 4^\circ
Step 2: Simplify the left side.
\frac{360}{n(n+1)} = 4 \Rightarrow n(n+1) = 90
Step 3: Solve.
n = 9
Final answer: n = 9.
The common mistake here is forgetting that interior and exterior angles at a vertex add to 180°. Use that relation whenever the question gives one angle and asks for the other.
14.4 QUADRILATERALS
A quadrilateral is a closed plane figure bounded by four line segments. The segments are its sides, and their endpoints are its vertices. The segments joining opposite vertices are its diagonals. A quadrilateral ABCD has two diagonals, AC and BD.
The sum of the angles of a quadrilateral is 360°. This is the polygon formula with n = 4: (2 \times 4 - 4) \times 90^\circ = 360^\circ. A diagonal splits a quadrilateral into two triangles, and each triangle contributes 180°, which is why the total comes to 360°.
This short section sets up the special quadrilaterals that follow. Each one is a quadrilateral with extra conditions on its sides, angles or diagonals.
14.5 SPECIAL KINDS OF QUADRILATERALS
Trapezium
A trapezium is a quadrilateral in which one pair of opposite sides is parallel and the other pair is non-parallel. When the non-parallel sides are equal, it is an isosceles trapezium. In an isosceles trapezium the base angles are equal in pairs, the diagonals AC and BD are equal, and the diagonals cut each other so that OA = OB and OC = OD.
Parallelogram
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Its properties form the backbone of the chapter.
- Opposite sides are parallel and equal.
- Opposite angles are equal.
- Consecutive angles are supplementary.
- Diagonals bisect each other.
- Each diagonal divides it into two congruent triangles.
- The diagonals divide it into four triangles of equal area.
Rectangle
A rectangle is a parallelogram in which every angle is 90°. Its diagonals are equal and bisect each other. Each diagonal divides the rectangle into two congruent triangles.
Rhombus
A rhombus is a parallelogram in which all four sides are equal. Its diagonals bisect each other at 90°, and each diagonal bisects the angles at its vertices.
Square
A square is a parallelogram with all sides equal and every angle 90°. Its diagonals are equal, bisect each other at right angles, and bisect the vertex angles. A square is the only shape that combines every property in the table below.
| Property | Parallelogram | Rectangle | Rhombus | Square |
|---|---|---|---|---|
| Diagonals bisect each other | Yes | Yes | Yes | Yes |
| Diagonals are equal | No | Yes | No | Yes |
| Diagonals bisect vertex angles | No | No | Yes | Yes |
| Diagonals are perpendicular | No | No | Yes | Yes |
| Each diagonal forms 2 congruent triangles | Yes | Yes | Yes | Yes |
| Diagonals form 4 congruent triangles | No | No | Yes | Yes |
The chapter proves seven theorems about these shapes. Theorem 11 shows opposite sides of a parallelogram are equal by ASA on triangles ABC and CDA. Theorem 12 shows opposite angles are equal. Theorem 13 proves that one pair of equal and parallel opposite sides makes a quadrilateral a parallelogram.
Theorem 14 states each diagonal bisects the parallelogram. Theorem 15 proves the diagonals bisect each other. Theorem 16 proves the diagonals of a rhombus meet at 90° using SSS. Theorem 17 proves the diagonals of a rectangle are equal using SAS.
Worked pattern: in parallelogram ABCD, AB = 4x − 2 and DC = 3x + 5. Find x and the length of AB.
Step 1: Opposite sides of a parallelogram are equal.
4x - 2 = 3x + 5
Step 2: Solve for x.
x = 7
Step 3: Find AB.
AB = 4(7) - 2 = 26
Final answer: x = 7 and AB = 26 units.
Common mistakes in this section: assuming a rhombus is a square, or a rectangle is a square. A rhombus has equal sides but not necessarily right angles. A rectangle has right angles but not necessarily equal sides. Only a square has both. Remember the mnemonic: all sides equal in a rhombus, all angles 90° in a rectangle, both in a square.
These theorems build directly into the next chapter on construction of polygons, available in the Class 9 book list.
What to practise from Rectilinear Figure Class 9 ICSE
Work through the three exercises in order. Each builds on the skills of the previous one.
- EXERCISE 14(A) — 11 questions on polygon names and angle sums.
- EXERCISE 14(B) — 10 questions on special quadrilaterals and their properties.
- EXERCISE 14(C) — 19 questions on proofs involving parallelograms and related shapes.
- “Prove the following” set inside section 14.5 — 11 statements that apply the theorems in reverse.
Working through all three in order gives you practice in every skill this chapter introduces. The official CISCE website lists the syllabus and examination details.
Rectilinear Figure Class 9 ICSE frequently asked questions
What is a rectilinear figure in Class 9 ICSE Maths?
A rectilinear figure is a plane figure bounded by straight lines. When the figure is closed and has at least three sides, it is called a polygon. Triangles, quadrilaterals and pentagons are all rectilinear figures.
How do I find the number of sides of a regular polygon in Class 9 ICSE?
Divide 360° by the exterior angle. If the exterior angle is x^\circ, the number of sides is \frac{360}{x}. For example, an exterior angle of 40° gives \frac{360}{40} = 9 sides.
What are the properties of a rectangle in Class 9 ICSE?
A rectangle is a parallelogram with all angles equal to 90°. Its diagonals are equal and bisect each other. Each diagonal divides the rectangle into two congruent triangles.
How do I prove that the diagonals of a parallelogram bisect each other in Class 9 ICSE?
This is Theorem 15. Draw diagonals AC and BD meeting at O. Prove triangles AOB and COD congruent by ASA, using equal opposite sides and alternate angles. Then OA = OC and OB = OD by CPCTC.
What is the sum of interior angles of a pentagon in Class 9 ICSE Maths?
A pentagon has 5 sides, so use the formula (2n - 4) \times 90^\circ. That gives (2 \times 5 - 4) \times 90^\circ = 540^\circ. Each angle of a regular pentagon is 108°.
Reference: ICSE Class 9 Mathematics textbook material, chapter 14: Rectilinear Figures — CISCE-aligned.