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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCylinderDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremScalene triangleSetsSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers

Types of angles

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Understanding Types of Angles: A Comprehensive Guide

Comprehensive Definition, Description, Examples & Rules 

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What are the types of angles, angle names, and the difference between obtuse and acute???? Are these questions troubling you? Well, then let Edulyte  put your worries to rest. With these resources, you get exclusive tricks to understand angles and learn to handle angle calculations easily!

Introduction: Angles In Our Daily Lives

Did you know that the importance of understanding types of angles in geometry extends far beyond the classroom?

  • Angles in geometry provide a framework for measuring and describing the orientation of lines and shapes, enabling mathematicians to analyse and solve complex problems related to space and dimension.
  • In physics, angles are integral to the study of motion, forces, and equilibrium. Concepts such as projectile motion, vectors, and the principles of mechanics heavily rely on an understanding of angles.
  • In engineering, different types of angles are essential in the design and construction of structures. Civil engineers, architects, and mechanical engineers, among others, use angles to ensure the integrity and functionality of their designs.

Fundamentals of Angles: Definition And Angles Names

Definition of Angles

In geometry, an angle forms when two rays or line segments share a common endpoint called the vertex. The rays extending from the vertex are the sides of the angle. Angles are measured in degrees.

Significance of Angles

The significance of types of angles extends from mathematics to physics, engineering, navigation, and various other disciplines where spatial relationships play a vital role.

Basic Classification of Types of Angles Based on Measurement

Acute Angle:

  • It measures less than 90 degrees.
  • Example: A 45-degree angle is acute because it is less than 90 degrees.

Obtuse Angle:

  • It measures more than 90 degrees but less than 180 degrees.
  • Example: A 120-degree angle is obtuse, greater than 90 degrees but less than 180 degrees.

Right Angle:

  • A right angle is exactly 90 degrees.
  • Example: The angle formed by the meeting of the adjacent sides of a square is a right angle.

Straight Angle:

  • A straight angle is exactly 180 degrees.
  • It forms a straight line, and the two rays or line segments constituting it are opposite and collinear.
  • Example: The angle formed by a straight line is straight.

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Common Angle Names

All types of angles are classified based on their measurements and characteristics. Here are some commonly used angle names:

Acute Angle:

  • It measures less than 90 degrees.
  • Examples: 30 degrees, 45 degrees, and 60 degrees are all acute angles.

 

Acuteangle

Obtuse Angle:

  • It measures more than 90 degrees but less than 180 degrees.
  • Examples: 100 degrees, 120 degrees, and 150 degrees are all obtuse angles.

Obtuse Angle

Right Angle:

  • A right angle is exactly 90 degrees.
  • It forms a perfect “L” shape.
  • Example: The angle formed by the corner of a square is a right angle.

Right Angle

Straight Angle:

  • A straight angle is exactly 180 degrees.
  • It forms a straight line, and the two rays or line segments constituting it are opposite and collinear.
  • Example: The angle formed by a straight line is straight.

Straight Angle

Reflex Angle:

  • It measures more than 180 degrees but less than 360 degrees.
  • It is formed by continuing one of the angle’s sides beyond 180 degrees.
  • Example: An angle measuring 220 degrees is a reflex angle.

Reflex Angle

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Types of Angles Based on Measurement

Angles can be classified into various types based on their measurements. Understanding these types is crucial in geometry. Here’s a detailed exploration of different angle measurements, focusing on acute angles, obtuse angles, and right angles:

Acute Angle:

  • It measures less than 90 degrees.
  • Acute angles are characterised by their sharpness and are commonly found in triangles and other geometric shapes.
  • Examples: A 30-degree angle, a 45-degree angle, and a 60-degree angle are all acute angles.

Obtuse Angle:

  • It measures more than 90 degrees but less than 180 degrees.
  • Obtuse angles are broader and less sharp than acute angles.
  • Examples: A 100-degree angle, a 120-degree angle, and a 150-degree angle are all obtuse angles.

Right Angle:

  • A right angle is exactly 90 degrees.
  • It forms a perfect “L” shape, resembling the corner of a square or rectangle.
  • Right angles are crucial in geometric constructions and are often used as a reference point for measuring and constructing other angles.
  • Example: The angle formed by the intersection of two perpendicular lines is right.

Special Types of Angles

In addition to the primary classification of angles based on measurement, there are special types of angles that exhibit specific relationships with one another.

Complementary Angles:

  • Sum of complementary angles is precisely 90 degrees.
  • In other words, when you add the measure of one angle to the measure of another angle, the result is a right angle (90 degrees).
  • Example: If one angle measures 45 degrees, its complementary angle would measure 45 degrees as well (45 degrees + 45 degrees = 90 degrees).

Complementary Angles

Supplementary Angles:

  • Two angles are supplementary when their sum is exactly 180 degrees.
  • Adding the measure of one angle to the measure of another angle results in a straight line.
  • Example: If one angle measures 120 degrees, its supplementary angle would measure 60 degrees (120 degrees + 60 degrees = 180 degrees).

Supplementary Angles

Vertical Angles:

  • Vertical angles are pairs of angles formed by intersecting lines. They are opposite each other and share a common vertex.
  • Vertical angles are always congruent, meaning they have the same measure.
  • Example: If ∠A and ∠B are vertical angles in the intersecting lines, then ∠A = ∠B.

Vertical Angles

Polygons and Angles

Polygons are closed geometric figures with straight sides. The angles within polygons define their properties and characteristics.

Polygons and Angles

Interior Angles of a Polygon

  • Sum of interior angles of polygon = (where n is the number of sides) 180∘×(n−2).
  • In a regular polygon (a polygon with all sides and angles equal), every interior angle is the result of dividing the sum of the interior angles by the number of sides.

Example:

  • In a triangle (n=3), the sum of interior angles is 180∘×(3−2)=180∘. Therefore, each interior angle in a triangle is 180∘/3=60∘.

Exterior Angles of a Polygon

  • An exterior angle of polygon is formed between one side of the polygon and the extension of an adjacent side.
  • Sum of exterior angles of polygon, regardless of the number of sides, is always 360∘.

Example:

  • In a triangle, each exterior angle is supplementary to its corresponding interior angle. If one interior angle is 60∘, the corresponding exterior angle is 180∘−60∘=120∘.

Applications of Angles

  • Building Design and Construction: Architects use angles extensively in designing and constructing buildings. Right angles ensure structural stability, while acute and obtuse angles contribute to the aesthetics and functionality of architectural elements.
  • Structural Engineering: = Understanding the angles formed by beams, trusses, and other components is essential for ensuring the structural integrity and stability of bridges, buildings, and other infrastructure.
  • Mechanical Engineering: The angles of gears, cam profiles, and linkages are critical for achieving specific mechanical movements and functionalities.
  • Navigation: In navigation, understanding angles is fundamental for determining direction. Nautical charts, maps, and GPS systems rely on angles to provide accurate information about routes and locations.
  • Sports: In sports like golf, billiards, or archery, players consider the angles of impact or trajectory to achieve desired outcomes.
  • Home Improvement: DIY projects and tasks like cutting mouldings involve angles. An understanding of angles is required for measuring and cutting materials and installing fixtures.

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Key Takeaways

  1. Understanding angles is foundational to geometry.

  2. Angles are classified based on their measurements, with acute angles (less than 90 degrees), obtuse angles (more than 90 degrees but less than 180 degrees), and right angles (exactly 90 degrees) being common types.

  3. Special types of angles, such as complementary angles (sum to 90 degrees), supplementary angles (sum to 180 degrees), and vertical angles (opposite angles formed by intersecting lines), have specific properties that aid in problem-solving

  4. The sum of interior angles in a polygon is determined by the formula 180∘×(n−2), where n is the number of sides.

  5. Angles have practical applications in various fields, including architecture, engineering, navigation, and daily activities.

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Frequently Asked Questions

  • Two angles are complementary if they sum up to exactly 90 degrees.
  • Two angles are supplementary if they add up to exactly 180 degrees.

Angles are closely related to polygons; understanding this relationship is fundamental in geometry. Here are some critical aspects of how angles relate to polygons:

Interior Angles: Sum of interior angles of polygon formula Sum=(n−2)×180∘ (n = number of sides).

Exterior Angles: An exterior angle of a polygon forms between one side of the polygon and the extension of an adjacent side. Sum of exterior angles of polygon is always 360∘

Vertical angles are pairs of angles formed by the intersection of two lines. They share a common vertex but are not adjacent angles; instead, they are opposite each other. Vertical angles are always congruent, meaning they have the same measure.

Some of the ways to determine the measurement of an angle are:

Protractor: The most direct method is to use a protractor. Place the protractor’s centre over the angle’s vertex, align one side of the angle with the baseline of the protractor, and read the measurement where the other side intersects the protractor scale.

Known Angle Relationships: If the angle is part of a geometric figure with known angle relationships (e.g., a triangle or a straight line), you can use those relationships to find the angle. 

Complementary and Supplementary Angles: If you have information about complementary or supplementary angles, you can use that information to find the angle. 

Trigonometry: If you have information about the sides of a right-angled triangle, you can use trigonometric functions (sine, cosine, tangent) to find the angle. 

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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCylinderDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremScalene triangleSetsSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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