MathsICSE Class 9 · Rectilinear Figures

Angles of Polygons and Properties of a Parallelogram (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

The interior angles of an n-sided polygon add up to (n − 2) × 180° and its exterior angles (one at each vertex) add up to 360°. Each exterior angle of a regular polygon is 360°/n. In a parallelogram opposite sides and angles are equal, adjacent angles add up to 180°, and the diagonals bisect each other.

Understand Angles of Polygons and Properties of a Parallelogram

Polygon angles. For a polygon with $n$ sides:

Sum of interior angles $= (n - 2) \times 18 0^{\circ}$ (split into $n - 2$ triangles from one vertex).
Sum of exterior angles (one at each vertex) $= 36 0^{\circ}$ .
Each interior angle of a regular polygon $= \frac{( n - 2 ) \times 18 0 ^{\circ}}{n}$ ; each exterior angle $= \frac{36 0 ^{\circ}}{n}$ ; interior $+$ exterior $= 18 0^{\circ}$ .
Number of diagonals $= \frac{n ( n - 3 )}{2}$ .

So $n = \frac{36 0 ^{\circ}}{exterior angle}$ . A regular hexagon has interior sum $72 0^{\circ}$ ; a regular pentagon has each interior angle $10 8^{\circ}$ .

Parallelogram (prove). Opposite sides parallel. Properties: (i) opposite sides are equal; (ii) opposite angles are equal; (iii) adjacent angles are supplementary; (iv) the diagonals bisect each other; (v) each diagonal divides it into two congruent triangles.

*Proof of (i) and (ii).* Join diagonal $A C$ of $A B C D$ . Since $A B ∥ D C$ and $A D ∥ B C$ , alternate angles $∠ B A C = ∠ D C A$ and $∠ B C A = ∠ D A C$ . With $A C$ common, $△ A B C ≅ △ C D A$ (ASA), so $A B = C D$ , $B C = D A$ and $∠ B = ∠ D$ .

Converses (tests for a parallelogram). A quadrilateral is a parallelogram if: both pairs of opposite sides are equal; both pairs of opposite angles are equal; one pair of opposite sides is equal and parallel; or its diagonals bisect each other.

Setting up equations. If angles are given as algebraic expressions, use the right sum: adjacent angles of a parallelogram sum to $18 0^{\circ}$ , all four angles of a quadrilateral to $36 0^{\circ}$ , and diagonal halves are equal.

Level. Class IX is a school-exam class. Single-step angle questions are routine; the ratio, regular-polygon and diagonal-segment items are at the hardest school-exam level.

Angles of Polygons and Properties of a Parallelogram: formulas to remember

Interior angle sum: $(n - 2) \times 18 0^{\circ}$; n sides.
Exterior angle of regular polygon: $\frac{36 0 ^{\circ}}{n}$; Interior = 180° − exterior.
Diagonals: $\frac{n ( n - 3 )}{2}$; Of an n-gon.
Parallelogram angles: $∠ A + ∠ B = 18 0^{\circ}, ∠ A = ∠ C$; Adjacent supplementary.
Diagonals: $A O = O C, B O = O D$; They bisect each other.

How to solve Angles of Polygons and Properties of a Parallelogram questions

Choose the right sum (interior 360° for a quadrilateral, 180° for adjacent parallelogram angles), form an equation, solve, check.

1Identify the figure and the property you can use.
2Write an equation for the unknown (sum of angles, equal angles, equal halves of diagonals).
3Solve for the variable and substitute back to find each angle or length.
4Check that the total is 360° (or 180° for adjacent angles).
5For regular polygons, find $n$ from the exterior angle first.
6State the property used (for example "opposite angles of a parallelogram").

Angles of Polygons and Properties of a Parallelogram in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Tiles that fit around a point

Suppose a floor designer wants to use regular octagon tiles with small square tiles filling the gaps.

Show that two octagons and one square fit around a point.

Try it first, then show the solutionSolution

Interior angle of a regular octagon $= \frac{( 8 - 2 ) \times 180}{8} = 13 5^{\circ}$ .
$135 + 135 + 90 = 36 0^{\circ}$ , so they meet exactly around a point.

Answer: Yes: 135° + 135° + 90° = 360°.

Memory hook: Angles around a point add to 360°.

Bracing a gate

Imagine a parallelogram gate frame with one corner measuring $10 0^{\circ}$ .

Find all the angles.

Try it first, then show the solutionSolution

Opposite angles: $10 0^{\circ}$ and $10 0^{\circ}$ .
Adjacent angle: $18 0^{\circ} - 10 0^{\circ} = 8 0^{\circ}$ , and its opposite also $8 0^{\circ}$ .

Answer: 100°, 80°, 100°, 80°.

Memory hook: Adjacent angles add to 180°.

Tricks to remember Angles of Polygons and Properties of a Parallelogram

Short memory aids. Say each one aloud once, then use it on a question the same day.

n minus 2 triangles
A polygon splits into n − 2 triangles.
Use it when: Remembering the angle sum.
Exterior is 360 over n
Regular polygon: exterior = 360° ÷ n.
Use it when: Finding n.
Cross diagonals share a midpoint
Parallelogram diagonals bisect each other.
Use it when: Equations from diagonal pieces.

Angles of Polygons and Properties of a Parallelogram: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Interior angle sum · 1 mark · MCQBoard level

The sum of the interior angles of a hexagon is:
(a) $54 0^{\circ}$
(b) $72 0^{\circ}$
(c) $90 0^{\circ}$
(d) $108 0^{\circ}$

Try it first, then show the solutionSolution

$(6 - 2) \times 18 0^{\circ} = 72 0^{\circ}$ .

Answer: (b) $72 0^{\circ}$

Example 2: Regular polygon · 3 marks · Short answerBoard level

Each exterior angle of a regular polygon is $3 6^{\circ}$ . Find the number of sides, each interior angle and the sum of the interior angles.

Try it first, then show the solutionSolution

$n = \frac{360}{36} = 10$ .
Interior angle $= 180 - 36 = 14 4^{\circ}$ .
Sum $= (10 - 2) \times 180 = 144 0^{\circ}$ (also $10 \times 144 = 144 0^{\circ}$ ).

Answer: 10 sides; $14 4^{\circ}$ ; $144 0^{\circ}$

Example 3: Angles in a ratio · 3 marks · Short answerHarder board level

The angles of a quadrilateral are in the ratio $2 : 3 : 4 : 6$ . Find each angle.

Try it first, then show the solutionSolution

Sum $= 36 0^{\circ}$ and $2 + 3 + 4 + 6 = 15$ parts, so one part $= 2 4^{\circ}$ .
Angles: $4 8^{\circ}, 7 2^{\circ}, 9 6^{\circ}, 14 4^{\circ}$ .
Check: $48 + 72 + 96 + 144 = 360$ . ✓

Answer: $4 8^{\circ}, 7 2^{\circ}, 9 6^{\circ}, 14 4^{\circ}$

Example 4: Parallelogram with algebraic angles · 4 marks · Short answerHarder board level

In parallelogram $P QR S$ , $∠ P = (5 x - 20)^{\circ}$ and $∠ Q = (3 x + 40)^{\circ}$ . Find $x$ and all four angles.

Try it first, then show the solutionSolution

Adjacent angles are supplementary: $5 x - 20 + 3 x + 40 = 180$ .
$8 x = 160$ , so $x = 20$ .
$∠ P = ∠ R = 8 0^{\circ}$ , $∠ Q = ∠ S = 10 0^{\circ}$ .

Answer: $x = 20$ ; $8 0^{\circ}, 10 0^{\circ}, 8 0^{\circ}, 10 0^{\circ}$

Example 5: A regular polygon from a ratio · 5 marks · Long answerHOTS / competency

In a regular polygon the ratio of an interior angle to an exterior angle is $8 : 1$ . Find the number of sides, the sum of the interior angles and the number of diagonals.

Try it first, then show the solutionSolution

Let the exterior angle be $x$ ; the interior is $8 x$ , and $8 x + x = 180$ , so $x = 2 0^{\circ}$ .
$n = \frac{360}{20} = 18$ .
Sum of interior angles $= (18 - 2) \times 180 = 288 0^{\circ}$ (check: $18 \times 160 = 2880$ ).
Diagonals $= \frac{18 \times 15}{2} = 135$ .

Answer: 18 sides; 2880°; 135 diagonals

Where students lose marks in Angles of Polygons and Properties of a Parallelogram

The slips examiners see most often, and what to write instead.

Mistake: Using 180 for all four angles
A quadrilateral sums to 360°.
Wrong working
$2 x + 3 x + 4 x + 6 x = 180$ .
Right working
$15 x = 360$ .
Do this: Count the sides.
Usually costs: Whole answer.
Mistake: Interior = 360/n
That is the exterior angle.
Wrong working
Each interior angle of a regular decagon is $3 6^{\circ}$ .
Right working
$180 - 36 = 14 4^{\circ}$ .
Do this: Subtract from 180°.
Usually costs: Whole answer.
Mistake: Opposite instead of adjacent
Adjacent angles are supplementary; opposite are equal.
Wrong working
$∠ A + ∠ C = 18 0^{\circ}$ .
Right working
$∠ A = ∠ C$ and $∠ A + ∠ B = 18 0^{\circ}$ .
Do this: Label adjacent vs opposite.
Usually costs: Whole answer.

Exam tips

Class IX is examined in school papers; parallelogram properties are proved, and converses are tested.
Give the property you used beside each angle.
Check totals: 360° for a quadrilateral, 180° for adjacent angles.
Draw the diagonals with equal tick marks on their halves.

Practice questions on Angles of Polygons and Properties of a Parallelogram

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Each interior angle of a regular pentagon is:
(a) $7 2^{\circ}$
(b) $10 8^{\circ}$
(c) $12 0^{\circ}$
(d) $13 5^{\circ}$
Hint
$\frac{3 \times 180}{5}$ .
Answer
(b) $10 8^{\circ}$
Q2 · 3 marks · Short answerHarder board level
Two adjacent angles of a parallelogram are in the ratio $2 : 3$ . Find all its angles.
Hint
They add to 180°.
Answer
One part $= 3 6^{\circ}$ : $7 2^{\circ}$ and $10 8^{\circ}$ ; the angles are $7 2^{\circ}, 10 8^{\circ}, 7 2^{\circ}, 10 8^{\circ}$ .
Q3 · 4 marks · Short answerHarder board level
The diagonals $A C$ and $B D$ of parallelogram $A B C D$ meet at $O$ . $A O = (2 x + 1)$ cm, $O C = (3 x - 4)$ cm, $B O = (y + 2)$ cm and $O D = (2 y - 3)$ cm. Find $A C$ and $B D$ .
Hint
AO = OC and BO = OD.
Answer
$2 x + 1 = 3 x - 4 \Rightarrow x = 5$ , $A O = 11$ , $A C = 22$ cm. $y + 2 = 2 y - 3 \Rightarrow y = 5$ , $B O = 7$ , $B D = 14$ cm.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): A quadrilateral with both pairs of opposite angles equal is a parallelogram.
Reason (R): If $∠ A = ∠ C$ and $∠ B = ∠ D$ , then $∠ A + ∠ B = 18 0^{\circ}$ , making $A D ∥ B C$ and $A B ∥ D C$ .
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
$2 (A + B) = 360$ .
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A hall floor uses regular twelve-sided (dodecagonal) tiles with small equilateral triangular tiles filling the gaps.
(a) Find each interior angle of a regular dodecagon.
(b) Find each exterior angle.
(c) Show that two dodecagons and one triangle fit exactly around a point.
(d) Find the sum of the interior angles of one dodecagon.
Hint
Around a point: 360°.
Answer
(a) $15 0^{\circ}$ (b) $3 0^{\circ}$ (c) $150 + 150 + 60 = 360$ (d) $180 0^{\circ}$

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Angle sum of an n-gon?Show answer
(n − 2) × 180°.
2.Exterior angle of a regular n-gon?Show answer
360°/n.
3.Adjacent angles of a parallelogram?Show answer
Supplementary.
4.What do the diagonals of a parallelogram do?Show answer
They bisect each other.

Angles of Polygons and Properties of a Parallelogram: frequently asked questions

Are the converses of the parallelogram properties tested?

Yes, as ways to prove a quadrilateral is a parallelogram.

Does the exterior sum depend on n?

No, it is 360° for every convex polygon.

What is a regular polygon?

One with all sides and all angles equal.