Quadrilaterals Class 8 Notes: Properties, Types & Solved Examples

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What Is a Quadrilateral?

The name itself explains it: "Quad" means four, and "lateral" means side. So, a figure made of four straight sides is called a quadrilateral. It is a two-dimensional, closed figure with four sides, four vertices, and four angles.

Quadrilaterals belong to the larger family of polygons, which you may already know from earlier classes:

Diagonals, Regular and Irregular Polygons 

A diagonal is a line joining two non-consecutive (non-adjacent) vertices of a polygon. For example, in quadrilateral ABCD, joining A and C, or B and D, gives you a diagonal — because these pairs of vertices are not next to each other.

This applies even to a concave quadrilateral: even in a "pushed-in" shape, joining the non-adjacent vertices (such as B and D) still gives a valid diagonal.

Regular vs Irregular Polygons

• A regular polygon has all sides and all angles equal — for example, an equilateral triangle, a square (regular quadrilateral), or a regular pentagon.
• An irregular polygon has sides and/or angles that are not all equal — for example, a rectangle, or a hexagon or pentagon where only some sides are equal.

Angle Sum Property of a Quadrilateral 

You may already know that the angle sum property of a triangle is 180°. This is exactly where the quadrilateral's angle sum property comes from.

If you draw one diagonal in a quadrilateral, it splits the shape into two triangles. Each triangle has an angle sum of 180°, so together:

Angle A + Angle B + Angle C + Angle D = 360°

💡 Quick Recall: Two triangles = 360°, three triangles = 540°, and so on — the angle sum keeps increasing by 180° for every extra triangle a polygon can be split into.

Solved Example 1: Find the Missing Angle

Question: In a quadrilateral, Angle A = 110°, Angle B = 70°, Angle C = 80°. Find Angle D.

Solution: Angle A + Angle B + Angle C + Angle D = 360° 110° + 70° + 80° + Angle D = 360° 260° + Angle D = 360° Angle D = 360° − 260° = 100°

Solved Example 2: Angles in a Given Ratio

Question: The interior angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles.

Solution: Let the angles be 3x, 5x, 9x, and 13x. 3x + 5x + 9x + 13x = 360° 30x = 360° x = 12°

So:

• Angle A = 3 × 12 = 36°
• Angle B = 5 × 12 = 60°
• Angle C = 9 × 12 = 108°
• Angle D = 13 × 12 = 156°

(Check: 36 + 60 + 108 + 156 = 360° ✓)

Sum of Exterior Angles

As an additional, exam-relevant note beyond the textbook: the sum of the exterior angles of any polygon is always 360°. For example, if three exterior angles of a quadrilateral are 50°, 110°, and 90°, and the fourth is x:

50° + 110° + x + 90° = 360° 250° + x = 360° x = 110°

The Parallelogram: Properties and Solved Examples 

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel and equal.

Key properties:

1. Opposite sides are parallel and equal: AB = CD and BC = AD, and AB ∥ CD, BC ∥ AD
2. Opposite angles are equal: Angle A = Angle C, Angle B = Angle D
3. Adjacent angles are supplementary: Angle A + Angle B = 180° (and similarly for every other adjacent pair)
4. Diagonals bisect each other: AO = OC and BO = OD (where O is the point of intersection)
5. Each diagonal divides the parallelogram into two congruent triangles — this can be proved using the SSS congruence criterion, since both pairs of opposite sides are equal and the diagonal is common to both triangles

Solved Example 3: One Angle Given

Question: In parallelogram PQRS, Angle R = 70°. Find all the angles.

Solution:

• Angle P = Angle R = 70° (opposite angles are equal)
• Angle P + Angle Q = 180° (adjacent angles are supplementary) → Angle Q = 180° − 70° = 110°
• Angle Q = Angle S = 110° (opposite angles are equal)

Solved Example 4: Adjacent Angles in a Ratio

Question: Two adjacent interior angles of a parallelogram are in the ratio 4 : 5. Find both angles.

Solution: Let the angles be 4x and 5x. 4x + 5x = 180° (adjacent angles are supplementary) 9x = 180° x = 20°

So the angles are 4 × 20° = 80° and 5 × 20° = 100°.

Solved Example 5: Sum of Two Opposite Angles Given

Question: The sum of two opposite angles of a parallelogram is 130°. Find all four angles.

💡 Exam Tip: This exact question format appears very frequently in Class 8 school exam papers on this chapter — it is worth practising until it becomes automatic.

Solution: Since opposite angles are equal, if Angle A + Angle C = 130° and Angle A = Angle C: 2 × Angle A = 130° → Angle A = 65° = Angle C Angle B = 180° − 65° = 115° = Angle D

So: Angle A = 65°, Angle B = 115°, Angle C = 65°, Angle D = 115°.

The Rectangle: Properties and Solved Examples 

A rectangle is a special type of parallelogram whose angles are all right angles (90°).

Key properties:

1. Opposite sides are parallel and equal (same as a parallelogram)
2. All angles are equal to 90°
3. Diagonals are equal and bisect each other: AC = BD, and AO = OC, BO = OD

This can be proved by showing triangle ABD is congruent to triangle ABC using the SAS criterion (AD = BC, Angle A = Angle B = 90°, AB = AB common) — by CPCT, the diagonals AC and BD are equal.

Solved Example 6: Angles Formed by Diagonals

Question: ABCD is a rectangle with diagonals intersecting at O. Angle ABD = 30°. Find Angle ACD and Angle BDC.

Solution: Since the diagonals of a rectangle are equal and bisect each other, the triangles formed by the diagonals are isosceles. Angle ABD + Angle DBC = 90° (since Angle ABC = 90°), so Angle DBC = 60°. Because the relevant sides are equal, Angle BDC also works out to 30°, and by the same logic, Angle ACD = 30° as well.

Solved Example 7: Finding x Using Diagonal Lengths

Question: RENT is a rectangle whose diagonals meet at O. OR = 2x + 4 and OT = 3x + 1. Find x.

Solution: Since the diagonals of a rectangle are equal and bisect each other, OR = OT. 2x + 4 = 3x + 1 4 − 1 = 3x − 2x x = 3

Substituting back: OR = 2(3) + 4 = 10, and OT = 3(3) + 1 = 10 — both match, confirming the answer.

The Square: Properties 

A square is a rectangle in which a pair of adjacent sides are equal — which makes all four sides equal.

Key properties:

1. All sides and all angles are equal
2. Opposite sides are parallel
3. Diagonals are equal and bisect each other at right angles (90°)

A square is described as a special type of parallelogram whose all angles and all sides are equal. In a Venn diagram, a square lies inside the rectangle group, since every square is also a rectangle, but not every rectangle is a square.

The Rhombus: Properties and a Solved Example 

A rhombus is a parallelogram in which all four sides are equal — its angles, unlike a rectangle's, are not necessarily 90°.

Key properties:

1. All sides are equal
2. Opposite angles are equal (Angle A = Angle C, Angle B = Angle D)
3. Diagonals bisect the angles at each vertex
4. Adjacent angles are supplementary
5. Diagonals are perpendicular to each other, but — unlike in a rectangle or square — the diagonals of a rhombus are not equal, because the shape is slanted rather than symmetric

💡 Concept Clarity: Diagonals are equal only in a rectangle and a square, because these shapes are symmetric. In a parallelogram and a rhombus, diagonals are not equal because the shape is slanted. However, diagonals bisect each other in all four — parallelogram, rectangle, square, and rhombus. Diagonals meet at 90° only in the square and the rhombus.

Solved Example 8: Finding the Side of a Rhombus

Question: RICE is a rhombus where the half-diagonals OC = 12 cm and OI = 5 cm. Find the side lengths x, y, z.

Solution: Since the diagonals of a rhombus are perpendicular, triangle OCI is a right-angled triangle. Using the Pythagoras theorem:

IC² = OC² + OI² IC² = 12² + 5² = 144 + 25 = 169 IC = 13 cm

Since all sides of a rhombus are equal, the side of the rhombus (z) = 13 cm. The half-diagonal values also give x = 5 cm and y = 12 cm.

The Kite: Properties 

A kite is a quadrilateral with two pairs of adjacent sides equal — note that this is different from a parallelogram, where opposite sides are equal.

Key properties:

1. Two pairs of adjacent sides are equal
2. Diagonals meet at right angles (90°)
3. Exactly one pair of opposite angles is equal (the pair between the two unequal sides)

This last property can be shown by splitting the kite along its diagonal into two triangles: each pair of equal adjacent sides creates equal base angles, and adding these equal angles from both triangles shows that only one pair of opposite angles in the kite is equal.

The Trapezium and Isosceles Trapezium 

A trapezium is a quadrilateral with exactly one pair of opposite sides parallel. For example, if AB ∥ CD, then AB and CD are called the bases of the trapezium.

Key properties:

• No sides, angles, or diagonals are necessarily equal
• If the non-parallel sides happen to be equal, the trapezium is called an isosceles trapezium

How All Quadrilaterals Are Related (Venn Diagram)

The chapter ends with a Venn diagram connecting all these shapes, and it is a common source of exam questions, including diagram-based ones:

• Trapezium is the largest, most general group among these (only one pair of sides needs to be parallel).
• Parallelogram sits inside this larger family, since a parallelogram has both pairs of sides parallel.
• Rectangle and Rhombus are both special types of parallelograms.
• Square lies in the overlap of rectangle and rhombus, since a square satisfies the properties of both — all angles equal to 90° (like a rectangle) and all sides equal (like a rhombus).
• Kite stands apart from the trapezium-parallelogram family, since a kite's defining property (two pairs of adjacent equal sides) is different from a parallelogram's (opposite sides equal) — though a square also satisfies the kite's condition.

True or False Practice Questions 

Practising these statements builds the exact logic needed for assertion-reason questions, which frequently appear in Class 8 exams from this chapter.

1. All rectangles are squares. → False (a rectangle only becomes a square if its adjacent sides are also equal)
2. A quadrilateral whose diagonals are perpendicular to each other must be a rhombus. → False (a square also has perpendicular diagonals)
3. All squares are rhombuses and also rectangles. → True
4. All squares are not a parallelogram. → False (every square is a parallelogram)
5. A quadrilateral in which all angles are equal is a rectangle. → Considered False here, since a square (where all angles are also equal) is not specifically named
6. Isosceles trapeziums are a parallelogram. → False
7. A quadrilateral whose diagonals are equal and bisect each other must be a square. → False (a rectangle also satisfies this condition)
8. All squares are trapeziums. → True (a square has at least one pair of parallel sides, satisfying the trapezium condition)

For students starting their JEE and NEET foundation early — right from Class 8 — eSaral's live, animation-based classes under the Shakti course and Vajra NEET course build on exactly this kind of concept clarity. For more chapter-wise practice, explore eSaral's NCERT Solutions for Class 8 and NCERT Solutions for Class 8 Maths.