Topics
Number System(Consolidating the Sense of Numberness)
Number System
Estimation
Ratio and Proportion
Algebra
Numbers in India and International System (With Comparison)
Geometry
Place Value
Mensuration
Natural Numbers and Whole Numbers (Including Patterns)
Data Handling
Negative Numbers and Integers
Number Line
HCF and LCM
Playing with Numbers
- Simplification of Brackets
- Finding Factors Using Rectangular Arrangements and Division
- Factors and Common Factors
- Multiples and Common Multiples
- Concept of Even and Odd Number
- Tests for Divisibility of Numbers
- Divisibility by 2
- Divisibility by 4
- Divisibility by 8
- Divisibility by 3
- Divisibility by 6
- Divisibility by 9
- Divisibility by 5
- Divisibility by 11
Sets
Ratio
Proportion (Including Word Problems)
Unitary Method
Fractions
- Concept of Fraction
- Types of Fractions
- Concept of Proper and Improper Fractions
- Concept of Mixed Fractions
- Like and Unlike Fraction
- Concept of Equivalent Fractions
- Conversion between Improper and Mixed fraction
- Conversion between Unlike and Like Fractions
- Simplest Form of a Fractions
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of Fraction
- Division of Fractions
- Using Operator 'Of' with Multiplication and Division
- BODMAS Rule
- Problems Based on Fraction
Decimal Fractions
Percent (Percentage)
Idea of Speed, Distance and Time
Fundamental Concepts
Fundamental Operations (Related to Algebraic Expressions)
Substitution (Including Use of Brackets as Grouping Symbols)
Framing Algebraic Expressions (Including Evaluation)
Simple (Linear) Equations (Including Word Problems)
Fundamental Concepts
Angles (With Their Types)
Properties of Angles and Lines (Including Parallel Lines)
Triangles (Including Types, Properties and Constructions)
Quadrilateral
Polygons
The Circle
Symmetry (Including Constructions on Symmetry)
Recognition of Solids
Perimeter and Area of Plane Figures
Data Handling (Including Pictograph and Bar Graph)
Mean and Median
- Introduction
- Interior and Exterior of a Quadrilateral
- Explanation: Sum of the Interior Angles of a Quadrilateral
- Example 1
- Example 2
- Key Points Summary
Introduction
Have you ever noticed how many things around you have four sides? Your phone screen, a book, a football field, or the face of a brick! All these shapes are called quadrilaterals.
Understanding the properties of these shapes is crucial in fields like architecture, engineering, and even computer graphics. The most important rule about any closed, four-sided shape is what all its interior angles add up to. This is called the Angle Sum Property of a Quadrilateral.
Interior and Exterior of a Quadrilateral
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The interior of a quadrilateral is the region enclosed by its sides.
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The exterior of a quadrilateral is the space outside the figure.
- The shaded portion of the figure is called interior of the quadrilateral ABCD.
- the part of the plane which is outside the quadrilateral ABCD is called exterior of quadrilateral ABCD.
Explanation: Sum of the Interior Angles of a Quadrilateral
Step 1: Draw a Quadrilateral
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It has four sides and four angles: ∠A, ∠B, ∠C, and ∠D.
Step 2: Diagonal Divides the Quadrilateral
- The diagonal BD divides the quadrilateral into two triangles:
△ABD and △BDC
Step 3: Angles in Each Triangle
- We know that the sum of the angles of a triangle is always 180°.
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In △ABD → ∠DAB + ∠ABD + ∠ADB = 180°
In △BDC → ∠DBC + ∠BCD + ∠BDC = 180°
Step 4: Combine the Angles
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∠DAB + ∠ABC + ∠BCD + ∠CDA = 360°
Step 5: Conclusion
- Therefore, the sum of the measures of the four interior angles of any quadrilateral is 360°.
Example 1
Angles of a quadrilateral are 85°, 95°, x° and (x + 10)°. Find the value of x.
Solution:
Sum of the angles of the quadrilateral = 360°
⇒ 85° + 95° + x° + (x + 10)° = 360°
⇒ 180° + x° + x° + 10° = 360°
⇒ 2x° + 190° = 360°
i.e., 2x° = 360° − 190° = 170°
∴ x° = `"170°"/"2"`
∴ x = 85
Example 2
Three angles of a quadrilateral are in the ratio 4 : 6 : 3. If the fourth angle is 100°, find the other three angles of the quadrilateral.
Solution:
Let the three angles be 4x, 6x and 3x
∴ 4x + 6x + 3x + 100° = 360°
⇒ 13x = 360° − 100° = 260° and
x = `"260°"/"13"` = 20°
∴ The other three angles are 4x, 6x and 3x
= 4 × 20°, 6 × 20° and 3 × 20°
= 80°, 120° and 60°
Key Points Summary
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The sum of its interior angles is always 360°.
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Drawing a diagonal divides it into two triangles.
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Each triangle has a sum of angles = 180° → Total = 360°.
