Quadrilaterals
This page brings together the properties of quadrilaterals (parallelogram, rectangle, square, rhombus, trapezium), polygon angle formulas, and solved examples for REET. The Geometry section brings a question testing property recognition or angle calculation from this topic almost every year.
- 1 question in REET 2024
- Class 8 level
- 8 min read
How REET asks this topic
REET 2024 asked a question on finding the number of sides of a regular polygon from the ratio of its interior to exterior angle — showing that applying the polygon angle formulas in practice matters here, not just remembering definitions.
Basic Properties of a Quadrilateral
A quadrilateral is a polygon with four sides, and the sum of its four interior angles is always 360°.
For a polygon with n sides, the sum of interior angles = (n−2) × 180°, and the sum of its exterior angles is always 360°, no matter how many sides it has. In a regular polygon, each exterior angle = 360°/n and each interior angle = 180° − (360°/n).
Properties of Common Quadrilaterals
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Parallelogram | Opposite sides equal and parallel | Opposite angles equal; adjacent angles supplementary (sum 180°) | Bisect each other |
| Rectangle | Opposite sides equal and parallel | All angles 90° | Equal, and bisect each other |
| Rhombus | All four sides equal | Opposite angles equal; adjacent angles supplementary | Bisect each other perpendicularly (not equal in length) |
| Square | All four sides equal | All angles 90° | Equal, perpendicular, and bisect each other |
| Trapezium | One pair of opposite sides parallel | Co-interior angles on the same side sum to 180° | No special property |
Other Useful Polygon Formulas
- Number of diagonals in an n-sided polygon
- n(n−3)/2
- Triangles formed by splitting an n-sided polygon
- (n−2) — this is exactly where the interior angle sum formula (n−2)×180° comes from
- Each interior angle of a regular polygon
- 180° − (360°/n)
Properties of a Kite
In a kite, two pairs of adjacent sides are equal (not opposite ones), and one diagonal bisects the other perpendicularly, although the two diagonals don't divide each other into equal parts. One pair of its opposite angles (the ones between the unequal sides, i.e. where sides of different lengths meet) is equal, while the other pair (between the equal sides) can be unequal. True to its name, the shape of a flying kite is itself the most familiar example of this quadrilateral, which gives a natural, relatable reference when teaching it in class.
Solved Examples
Example 1 (PYQ-style): In a regular polygon, each exterior angle is 2/7 of the interior angle. Find the number of sides. Solution: Let the interior angle = I, so exterior angle = 2I/7. I + 2I/7 = 180° ⇒ 9I/7 = 180° ⇒ I = 140°, exterior angle = 40°. Sides = 360°/40° = 9.
Example 2: One angle of a parallelogram is 65°. Find the other three angles. Solution: Opposite angles are equal, so the opposite angle is also 65°. Adjacent angles are supplementary: 180°−65°=115°. Angles = 65°, 115°, 65°, 115°.
Example 3: A rectangle's diagonal is 26 cm and one side is 10 cm. Find the other side. Solution: By the Pythagoras theorem, diagonal² = side₁² + side₂² ⇒ 26² = 10² + b² ⇒ 676 = 100 + b² ⇒ b² = 576 ⇒ b = 24 cm.
Example 4: In a regular polygon, each interior angle is 5 times its exterior angle. Find the number of sides. Solution: Let exterior angle = E, interior angle = 5E. E + 5E = 180° ⇒ 6E = 180° ⇒ E = 30°. Sides = 360°/30° = 12.
How to Teach This to Children
Classroom activity
Cut out different quadrilaterals from paper and have students fold them along the diagonals — they'll see for themselves which ones have equal diagonals, which have perpendicular ones, and which have both. Geoboards and rubber bands can also be used to build various quadrilaterals and check their properties.
Teaching aids
A geoboard, dot-grid paper, and a quadrilateral properties chart are useful. Models built from straws and thread are especially helpful for showing diagonal properties (equal, perpendicular, bisecting).
Common misconception
Students often assume the square, rectangle and rhombus are three completely separate shapes, when in reality a square is also a special rectangle (with all sides equal) and also a special rhombus (with all angles 90°). Explaining this hierarchy is important.
Assessment idea
Build a yes/no table of properties for different quadrilaterals and ask which property applies to which shape — this checks classification understanding. Getting students to identify quadrilateral types in local objects (a window, a kite, a floor tile) is also a good assessment task.
Memory trick / exam tip
To quickly find the number of sides in questions about regular polygons, remember: sides = 360° ÷ exterior angle. If the question gives the interior angle, first find the exterior angle = 180° − interior angle, then apply the same formula — this saves time.
REET 2024 Questions (with Solutions)
Question 120
If each exterior angle of a regular polygon is 2/7 of its interior angle, then its number of sides will be
- A10
- B12
- C8
- D9
Explanation
बहिष्कोण + अन्त:कोण = 180°। यदि अन्त:कोण = I हो तो बहिष्कोण = 2I/7, अतः I + 2I/7 = 180° ⇒ 9I/7 = 180° ⇒ I = 140°, बहिष्कोण = 40°। समबहुभुज की भुजाओं की संख्या = 360°/बहिष्कोण = 360/40 = 9।
Practice Questions
Question 1
What is the sum of the interior angles of a quadrilateral?
- A360°
- B180°
- C270°
- D540°
Explanation
The four interior angles of any quadrilateral always add up to 360° (using the formula (n−2)×180° with n=4 gives 2×180°=360°).
Question 2
What is each interior angle of a regular hexagon?
- A120°
- B108°
- C135°
- D150°
Explanation
Interior angle = (n−2)×180°/n = (6−2)×180°/6 = 720°/6 = 120°.
Question 3
In a parallelogram, one angle is 65°. What is its adjacent angle?
- A115°
- B65°
- C25°
- D155°
Explanation
In a parallelogram, adjacent angles are supplementary: 180°−65°=115°.
Question 4
In which quadrilateral do the diagonals bisect each other at right angles but are NOT necessarily equal?
- Aसमचतुर्भुज
- Bआयत
- Cसमलंब
- Dकेवल वर्ग
Explanation
In a rhombus, the diagonals bisect each other perpendicularly, but are only equal in length when it is also a square. In a rectangle, the diagonals are equal but not perpendicular.
Question 5
Each exterior angle of a regular polygon is 45°. Find the number of sides.
- A8
- B9
- C6
- D10
Explanation
Sides = 360°/exterior angle = 360°/45° = 8.
Question 6
Each interior angle of a regular polygon is 5 times its exterior angle. Find the number of sides.
- A12
- B10
- C6
- D8
Explanation
Let exterior angle = E, interior angle = 5E. E+5E=180° ⇒ 6E=180° ⇒ E=30°. Sides = 360°/30° = 12.
Question 7
In trapezium ABCD, AB is parallel to CD. If ∠A=110°, what is ∠D?
- A70°
- B110°
- C90°
- D55°
Explanation
Since AB∥CD, the co-interior angles on the same side (AD) sum to 180°: ∠A+∠D=180° ⇒ ∠D=70°.
Question 8
The diagonal of a rectangle is 26 cm and one side is 10 cm. Find the length of the other side.
- A24 सेमी
- B20 सेमी
- C16 सेमी
- D26 सेमी
Explanation
By the Pythagoras theorem: 26² = 10² + b² ⇒ 676 = 100 + b² ⇒ b² = 576 ⇒ b = 24 cm.
Question 9
To explain that 'every square is also a rectangle', the most suitable classroom activity is:
- Aविभिन्न आकृतियों को उनके गुणों की तालिका बनाकर वर्गीकृत करना
- Bकेवल परिभाषा रटवाना
- Cचतुर्भुजों के नाम सूचीबद्ध कराना
- Dकोई गतिविधि आवश्यक नहीं
Explanation
A properties table lets children see for themselves that a square satisfies all of a rectangle's conditions (opposite sides equal-parallel, all angles 90°), building an understanding of the hierarchy.
Question 10
How many diagonals does a quadrilateral have in total?
- A2
- B4
- C1
- D3
Explanation
Number of diagonals in an n-sided polygon = n(n−3)/2. For n=4, this gives 4×1/2 = 2.
Frequently Asked Questions
How many questions come from quadrilaterals in REET?+
Usually 1 question from this topic appears — based on recognising properties or the polygon angle formula; REET 2024 also had 1 question.
What is the sum of the interior angles of a quadrilateral?+
The four interior angles of any quadrilateral always add up to 360°.
What's the difference between a rhombus and a square?+
A rhombus has all four sides equal, but its angles don't have to be 90°; a square has all four sides equal and all angles 90° — so a square is a special kind of rhombus.
Is every square also a rectangle?+
Yes, all of a square's features satisfy a rectangle's definition (opposite sides equal-parallel, all angles 90°), so a square is also a special type of rectangle.
What is the sum of the exterior angles of a polygon?+
No matter how many sides it has, the sum of all exterior angles of any convex polygon is always 360°.
What's the special feature of a kite?+
A kite has two pairs of equal adjacent sides, and its diagonals intersect perpendicularly, but they don't divide each other into equal parts.
How do you find the number of diagonals in an n-sided polygon?+
Using the formula n(n−3)/2, where n is the number of sides — for example, a pentagon (n=5) has 5×2/2 = 5 diagonals.
What's the most recognisable feature of a trapezium?+
A trapezium has only one pair of opposite sides parallel, while a parallelogram has both pairs of opposite sides parallel — this is the main difference between them.
What property do a parallelogram's diagonals satisfy?+
The two diagonals of a parallelogram always bisect each other, meaning the point of intersection divides both diagonals into two equal parts each.
In which quadrilateral do all three diagonal properties (equal, perpendicular, bisecting) occur together?+
Only in a square do all three of these properties occur together — its diagonals are equal, bisect each other, and also cut at right angles.
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Change Log
- 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked