# Chapter 10 – Geometry

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Geometry is an integral part of the mathematics syllabus from class 7 onwards. It is a branch of mathematics that deals with shapes and angles. It is the compilation of various methods you can use to measure the length, breadth, areas and angles subtended by these lines and shapes on a 2D or 3D plane.

#### Lines

A line in geometry is a one-dimensional figure with length but no breadth. It is formed when many points are extended linearly in opposite directions to any certain length extending upto infinity. On a two-dimensional plane, a line is defined by two points at the ends. These two points are also known as collinear points and lie on the same straight line.

In geometry, there are various kinds of lines, namely, horizontal lines, vertical lines, parallel lines and lines which are perpendicular to one another. These can be understood as the fundamental units of constructing any shape or polygon in geometry. For instance, a rectangle is made by joining four lines at the ends while a triangle is formed by three lines meeting at the ends.

#### Angles

In geometry, an angle is formed when two straight lines called rays or arms, are joined at a common point known as the vertex. The word angle is derived from the Latin word “Angulus”. The measure of angle is usually provided in degrees or radians. You can measure this using a protractor or a compass. They are generally denoted using Greek letters such as θ, α, β, and have three letter names where the centre mnemonic is for the vertex and the other two are for the arms of the angle formed.

Types of Angles

There are six main types of angles in mathematics. These have unique properties that are easy to understand. The list is as follows:

• Acute Angle: Angles that lie between 0° to 90°.
• Obtuse Angle: Angles that lie between 90° to 180°
• Right Angle: An angle that is exactly equal to 90°
• Straight Angle: An angle that is exactly equal to 180°
• Reflex Angle: An angle that is greater than 180° but less than 360°
• Full Rotation: An angle that is exactly equal to 360°

#### Interior and Exterior Angles

In a closed figure formed by the joining of extended points or lines, in geometry, there are at least 3 vertices where an angle is subtended inside of the figure. This figure can be any type of a shape or polygon including a triangle, pentagon, soiree, rhombus, etc. In all such geometrical 2D figures, there are interior angles and exterior angles.

• An interior angle can be understood as the angles subtended by the arms of the figure or the lines inside the closed figure formed when the lines are joined to one another.
• Similarly, the exterior angle property can be defined as the angles that are formed outside the closed figure or shape, between any side and line formed after extending the lines of the adjacent sides of that vertex.

#### 2D Figures in Geometry

For NCERT examinations, there are various types of two-dimensional closed figures that are tested. These shapes include triangles, squares, rectangles, quadrilaterals, parallelograms among other types of polygons.

#### Triangles in Geometry

In mathematics, a triangle can be defined as a closed figure or polygon with three sides. This shape or polygon consists of three edges and three vertices as well. And, the sum of all the internal angles in any triangle is equal to 180 degrees. This important property is called the angle sum property of triangles. The construction of simple triangles also needs this concept to be cleared. Moreover, if MOP is a triangle, then it is denoted as ∆MOP, where M, O and P are the three vertices of the triangle.

Properties of Triangles

Here are a few of the basic properties of triangles in geometry:

1. The sum of the interior angles of a triangle is 180 degrees.
2. The sum of exterior angles of a triangle is 360 degrees.
3. The sum of consecutive pairs of angles, one interior and one exterior angle is 180 degrees.
4. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

#### Conclusion

Geometry covers a range of topics such as shapes and figures, construction of simple triangles, 3D figures, angles and their properties, lines, among others. It is one of the most important branches in mathematics and it is one of the most scoring as well.

#### FAQs

1. What do you mean by lines in geometry?
A line in geometry can be defined as a straight set of points that extend in opposite directions and has a definite length but no defined width or breadth. It is a one-dimensional figure.
2. What are angles in geometry?
In geometry, an angle is formed when two straight lines called rays or arms, are joined at a common point known as the vertex. The measure of angle is usually provided in degrees or radians. They are generally denoted using Greek letters such as θ, α, β.
3. What is an obtuse angle in geometry?
In geometry, an obtuse angle can be defined as an angle that measures greater than 90 degrees but is less than a straight angle or 180 degrees. For instance, 92 degrees, 175 degrees, 120 degrees are examples of obtuse angles.
4. What is an exterior angle in geometry?
An exterior angle can be defined as the angles that are formed outside the closed figure or shape. This angle is formed between any side of the closed figure and a line, formed after extending the lines of the adjacent sides of that vertex.
5. What is a triangle in geometry?
A triangle can be defined as a closed figure or regular polygon with three sides. This shape or polygon consists of three edges and three vertices as well. And, the sum of all the internal angles in any triangle is equal to 180 degrees.

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### High School Physics

• Alternating Current
• Atoms
• Communication Systems
• Current Electricity
• Dual nature of Radiation and Matter
• Electric Charges and Fields
• Electricity
• Electromagnetic Induction
• Electromagnetic Waves
• Electron Beams and Radioactivity
• Electrons and Photons
• Electrostatic Potential and Capacitance
• Fluid Pressure
• Force and Acceleration
• Force And Laws Of Motion
• Gravitation
• Internal Energy
• Kinetic Theory
• Law of motion
• Light – Reflection And Refraction
• Magnetic Effects Of Electric Current
• Magnetism and Matter
• Management Of Natural Resources
• Mechanical properties of Fluids
• Mechanical properties of Solids
• Motion
• Motion in a plane
• Motion in a straight line
• Moving Charges and Magnetism
• Nuclear Energy
• Nuclei
• Oscillations
• Our Environment
• Paths of Heat
• Physical world
• Ray optics and optical instruments
• Semiconductor Devices
• Semiconductor Electronics: Materials, Devices and Simple Circuits
• Simple Machines
• Sound
• Sources Of Energy
• Specific and Latent Heats
• Spherical Mirrors
• Static Electricity
• Systems of Particles and Rotational motion
• Thermal properties of matter
• Thermodynamics
• Units and Measurement
• Vectors, Scalar Quantities and Elementary Calculus
• Wave Optics
• Waves
• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
• Aliphatic and Aromatic Hydrocarbons
• Alkyl and Aryl Halides
• Amines
• Analytical Chemistry
• Atomic Structure
• Atoms And Molecules
• Basic concepts of Chemistry
• Biomolecules
• Carbon And Its Compounds
• Carboxylic acids and Acid Derivatives
• Chemical Bonding and Molecular Structures
• Chemical Energetics
• Chemical Equilibria
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• Chemical Reactions And Equations
• Chemical Reactions and Their Mechanisms
• Chemistry in Everyday Life
• Chemistry of p-Block elements
• Chemistry of Transition and Inner Transition
• Classification of Elements
• Coordination Compounds
• Cyanide, Isocyanide, Nitro compounds and Amines
• Electrochemistry
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• States of Matter
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• Study of Compounds
• Study of Gas Laws
• Study of Representative Elements
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• The d-block and f-block elements
• The Gaseous State
• The p-Block Elements
• The Periodic Table
• The s-Block Elements
• The Solid State
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• Absorption and Movement of Water in Plants
• Adolescent Issues
• Anatomy of Flowering Plants
• Animal Kingdom
• Bacteria and Fungi-Friends and Foe
• Biodiversity and Conservation
• Biofertilizers
• Biological Classification
• Biomedical Engineering
• Biomolecules
• Biotechnology and its Applications
• Biotic Community
• Body Fluids and Circulation
• Breathing and Exchange of Gases
• Cell – Unit of Life
• Cell Cycle and Cell Division
• Cell Division and Structure of Chromosomes
• Cell Reproduction
• Cellular Respiration
• Chemical Coordination and Integration
• Circulation
• Control And Coordination
• Crop Improvement
• Digestion and Absorption
• Diversity In Living Organisms
• Ecosystem
• Environmental Issues
• Excretory Products and their Elimination
• Flowering Plants
• Genes and Chromosomes
• Health and Diseases
• Health and Its Significance
• Heredity And Evolution
• Heredity and Variation
• How Do Organisms Reproduce?
• Human Diseases
• Human Eye And Colourful World
• Human Health and Disease
• Human Population
• Human Reproduction
• Hygiene
• Improvement In Food Resources
• Integumentary System- Skin
• Kingdom Fungi
• Kingdom Monera
• Kingdom Protista
• Life Processes
• Locomotion and Movement
• Microbes in Human Welfare
• Mineral Nutrition
• Molecular Basis of Inheritance
• Morphology of Flowering Plants
• Neural Control And Coordination
• Nutrition in Human Beings
• Organism and Population
• Photosynthesis
• Photosynthesis in Higher Plants
• Plant Growth and Development
• Plant Kingdom
• Pollination and Fertilization
• Pollution; Sources and its effects
• Principles of Inheritance and Variation
• Reproduction and Development in Angiosperms
• Reproduction in Organisms
• Reproductive Health
• Respiration in Human Beings
• Respiration in Plants
• Respiratory System
• Sexual Reproduction in Flowering Plants
• Strategies for Enhancement in Food Production
• Structural Organisation in Animals
• Structural Organisation of the Cell
• The Endocrine System
• The Fundamental Unit Of Life
• The Living World
• The Nervous System and Sense Organs
• Tissues
• Transpiration
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### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
• Algebra – Linear Inequalities
• Algebra – Pair of Linear Equations in Two Variables
• Algebra – Polynomials
• Algebra – Principle of Mathematical Induction
• Algebra – Quadratic Equations
• Binomial Theorem
• Calculus – Applications of Derivatives
• Calculus – Applications of the Integrals
• Calculus – Continuity and Differentiability
• Calculus – Differential Equations
• Calculus – Integrals
• Geometry – Area
• Geometry – Circles
• Geometry – Conic Sections
• Geometry – Constructions
• Geometry – Introduction to Euclid’s Geometry
• Geometry – Three-dimensional Geometry
• Geometry – Lines and Angles
• Geometry – Quadrilaterals
• Geometry – Straight Lines
• Geometry – Triangles
• Linear Programming
• Matrices and Determinants
• Mensuration – Areas
• Mensuration – Surface Areas and Volumes
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• Number Systems – Real Numbers
• Permutations and Combinations
• Probability
• Sequence and Series
• Sets and Functions
• Statistics
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• Trignometry – Identities
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### Middle School Science

• Acids, Bases And Salts
• Air and Its Constituents
• Basic Biology
• Body Movements
• Carbon and Its Compounds
• Cell – Structure And Functions
• Changes Around Us
• Chemical Effects Of Electric Current
• Chemistry in Your Life
• Coal And Petroleum
• Combustion And Flame
• Components Of Food
• Conservation Of Plants And Animals
• Crop Production And Management
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• Electricity And Circuits
• Elements and Compounds
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• Garbage In, Garbage Out
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• Health and Hygiene
• Heat
• Hydrogen
• Life Processes: Nutrition in Animals and Plants
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• Materials: Metals And Non-Metals
• Matter and Its States
• Metals and Non-metals
• Micro Organisms: Friend And Foe
• Motion And Measurement Of Distances
• Motion And Time
• Nutrition In Animals
• Nutrition In Plants
• Organization in Living Things
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• Physical And Chemical Changes
• Pollution and conservation
• Pollution Of Air And Water
• Reaching The Age Of Adolescence
• Reproduction In Animals
• Reproduction In Plants
• Respiration In Organisms
• Rocks and Minerals
• Separation Of Substances
• Simple Machines
• Soil
• Some Natural Phenomena
• Sorting Materials Into Groups
• Sound
• Stars And The Solar System
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• Transformation of Substances
• Transportation In Animals And Plants
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• Water: A Precious Resource
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• Addition
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• Boxes and Sketches
• Data Handling
• Fun With Numbers
• Heavy and Light
• How Many
• Long And Short
• Mapping
• Measurement
• Money
• Multiplication and Factors
• Multiply and Divide
• Numbers
• Parts and Wholes
• Pattern Recognition
• Patterns
• Play With Patterns
• Rupees And Paise
• Shapes And Angles
• Shapes And Designs
• Shapes and Space
• Similarity
• Smart Charts
• Squares
• Subtraction
• Tables And Shares
• Tenths and Hundredths
• Time
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