Quadrilateral Basics
A quadrilateral is a closed shape made up of exactly four straight sides. The word comes from "quad" (meaning four) and "lateral" (meaning side).
Look around you — the top of your desk, the face of a book, a window pane, a floor tile. All of these are quadrilaterals!
In Class 6, you will learn about the parts of a quadrilateral — its sides, vertices, angles, and diagonals.
What is Quadrilateral Basics - Grade 6 Maths (Basic Geometrical Ideas)?
Definition: A quadrilateral is a closed figure formed by four line segments. It has:
- 4 sides (line segments)
- 4 vertices (corner points)
- 4 angles (at each vertex)
- 2 diagonals (lines joining opposite vertices)
A quadrilateral ABCD has:
- Sides: AB, BC, CD, DA
- Vertices: A, B, C, D
- Angles: ∠A, ∠B, ∠C, ∠D
- Diagonals: AC and BD
Quadrilateral Basics Formula
Key Properties of Every Quadrilateral:
- The sum of all four angles of a quadrilateral = 360°.
- A quadrilateral has exactly 2 diagonals.
∠A + ∠B + ∠C + ∠D = 360°
Adjacent and Opposite Sides:
- Adjacent sides share a common vertex. Example: AB and BC are adjacent (they meet at B).
- Opposite sides do not share a vertex. Example: AB and CD are opposite sides.
Adjacent and Opposite Angles:
- Adjacent angles share a common side. Example: ∠A and ∠B are adjacent (side AB is common).
- Opposite angles do not share a side. Example: ∠A and ∠C are opposite angles.
Types and Properties
Parts of a Quadrilateral:
- Interior: The region inside the quadrilateral.
- Exterior: The region outside the quadrilateral.
- Boundary: The four sides together form the boundary.
Convex and Concave Quadrilaterals:
- Convex quadrilateral: All interior angles are less than 180°. Both diagonals lie inside the quadrilateral.
- Concave quadrilateral: One interior angle is greater than 180°. One diagonal goes outside the quadrilateral.
Diagonal Facts:
- A diagonal divides a quadrilateral into two triangles.
- In quadrilateral ABCD, diagonal AC divides it into △ABC and △ACD.
- This is why the angle sum is 360° (two triangles × 180° = 360°).
Solved Examples
Example 1: Naming Parts of a Quadrilateral
Problem: In quadrilateral PQRS, name: (a) all sides, (b) all vertices, (c) all diagonals, (d) two pairs of opposite sides.
Solution:
- (a) Sides: PQ, QR, RS, SP
- (b) Vertices: P, Q, R, S
- (c) Diagonals: PR and QS
- (d) Opposite sides: PQ & RS, and QR & SP
Example 2: Finding Adjacent Sides
Problem: In quadrilateral ABCD, which sides are adjacent to side BC?
Solution:
Side BC has endpoints B and C.
- The side that shares vertex B with BC is AB.
- The side that shares vertex C with BC is CD.
Answer: AB and CD are adjacent to BC.
Example 3: Finding Opposite Angles
Problem: In quadrilateral WXYZ, name the pairs of opposite angles.
Solution:
- Pair 1: ∠W and ∠Y (they do not share a side)
- Pair 2: ∠X and ∠Z (they do not share a side)
Example 4: Angle Sum Property
Problem: Three angles of a quadrilateral are 90°, 85°, and 110°. Find the fourth angle.
Solution:
Sum of all angles = 360°
Fourth angle = 360° − (90° + 85° + 110°)
= 360° − 285° = 75°
Answer: The fourth angle is 75°.
Example 5: Angle Sum Property (Another Example)
Problem: The angles of a quadrilateral are x°, 2x°, 3x°, and 4x°. Find all the angles.
Solution:
x + 2x + 3x + 4x = 360°
10x = 360°
x = 36°
- First angle = 36°
- Second angle = 72°
- Third angle = 108°
- Fourth angle = 144°
Answer: The angles are 36°, 72°, 108°, and 144°.
Example 6: Counting Diagonals
Problem: How many diagonals does a quadrilateral have? Name them for quadrilateral KLMN.
Solution:
A quadrilateral always has 2 diagonals.
For KLMN, the diagonals are: KM and LN.
Example 7: Identifying Convex or Concave
Problem: A quadrilateral has angles 60°, 80°, 100°, and 120°. Is it convex or concave?
Solution:
All four angles are less than 180°.
Answer: It is a convex quadrilateral.
Example 8: Diagonal Divides into Triangles
Problem: In quadrilateral ABCD, diagonal BD is drawn. Name the two triangles formed.
Solution:
Diagonal BD divides ABCD into:
- △ABD
- △BCD
Real-World Applications
Where quadrilaterals appear in daily life:
- Buildings: Windows, doors, walls, and floors are all quadrilateral shapes.
- Books and screens: The face of your textbook, a TV screen, a mobile phone screen.
- Sports: A cricket pitch, a tennis court, a carrom board — all quadrilaterals.
- Art and design: Tiles, picture frames, and rangoli patterns use quadrilateral shapes.
- Maps: States and districts are often shown as approximate quadrilaterals.
Key Points to Remember
- A quadrilateral has 4 sides, 4 vertices, 4 angles, and 2 diagonals.
- The sum of all interior angles of a quadrilateral is 360°.
- Adjacent sides share a common vertex; opposite sides do not.
- Adjacent angles share a common side; opposite angles do not.
- A diagonal joins two opposite vertices.
- A diagonal divides a quadrilateral into two triangles.
- A convex quadrilateral has all angles less than 180°.
- A concave quadrilateral has one angle greater than 180°.
Practice Problems
- Name all sides, vertices, and diagonals of quadrilateral EFGH.
- Three angles of a quadrilateral are 70°, 100°, and 95°. Find the fourth angle.
- In quadrilateral ABCD, name the sides adjacent to AD.
- The angles of a quadrilateral are in the ratio 1:2:3:4. Find all the angles.
- Is it possible for a quadrilateral to have all four angles as 100°? Give a reason.
- Draw a quadrilateral and mark its two diagonals. How many triangles are formed?
Frequently Asked Questions
Q1. What is a quadrilateral?
A quadrilateral is a closed figure made of exactly four straight line segments. It has 4 sides, 4 vertices, and 4 angles. Examples include rectangles, squares, and parallelograms.
Q2. How many diagonals does a quadrilateral have?
A quadrilateral has exactly 2 diagonals. Each diagonal connects two opposite vertices. For example, in ABCD, the diagonals are AC and BD.
Q3. What is the angle sum property of a quadrilateral?
The sum of all four interior angles of any quadrilateral is always 360°. This is because a diagonal divides the quadrilateral into two triangles, and each triangle has an angle sum of 180°.
Q4. What are adjacent sides?
Adjacent sides are two sides of a quadrilateral that share a common vertex (corner point). In ABCD, AB and BC are adjacent because they meet at vertex B.
Q5. What is the difference between convex and concave quadrilaterals?
In a convex quadrilateral, all interior angles are less than 180° and both diagonals lie inside the shape. In a concave quadrilateral, one angle is greater than 180° and one diagonal goes outside the shape.
Q6. Is a triangle a quadrilateral?
No. A triangle has 3 sides, but a quadrilateral must have exactly 4 sides. They are different types of polygons.
Related Topics
- Introduction to Polygons
- Introduction to Triangles
- Types of Quadrilaterals
- Diagonals of a Polygon
- Point, Line Segment, Line and Ray
- Collinear and Non-Collinear Points
- Intersecting and Parallel Lines
- Curves - Open and Closed
- Circle - Basic Concepts
- Arc and Sector of a Circle
- Planes in Geometry
- Basic Geometrical Ideas
- Types of Lines
