Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Constructions ↓


Constructing Angles of Given Measure


In geometry, constructing angles is a basic skill that helps us understand the properties and principles that govern shapes and figures. Learning how to construct an angle of a given measure with accuracy requires an understanding of the basic tools and steps used in geometric constructions. In this detailed guide, we will discuss everything you need to know about constructing angles, focusing mainly on using a compass and straightedge.

Basic tools for angle construction

When plotting angles, the two most important tools are the compass and the straightedge. It is essential to be familiar with these tools:

  • Compass: This tool is used to draw arcs and circles. You can adjust the radius to help with accurate construction.
  • Straightedge: This helps you draw straight lines. Typically, a ruler is used, but for basic construction you don't need measurement markings.

A step-by-step guide to drawing angles

Let us learn how to construct different angles using compass and straight line.

Drawing a 60 degree angle

A 60 degree angle is one of the easiest angles to construct. Follow these steps:

  1. Draw a straight line using your straightedge. This will be one side of your angle. Label the starting point as A
    2. Line AB - side of the angle
  2. Open your compass to a convenient radius, place the compass point at A, and draw an arc crossing the line. Label the point where the arc intersects the line B.
    3. A B
  3. Without changing the width of the compass, place the compass at B and draw another arc that intersects the previous arc.
    4. A B C
  4. Label the point of intersection as C Now, use your straight line to draw a line from A to C
    5. Angle BAC is a 60-degree angle
    A B C

Constructing a 90 degree angle (right angle)

Constructing a 90 degree angle can be accomplished by constructing a perpendicular bisector:

  1. Start by drawing a straight line from your straight line and label the points as D and E
    2. Line DE - line to be bisected
  2. Adjust your compass to a width just over half that of DE. Place the compass point on D and draw arcs above and below the line.
    3. D I
  3. Without changing the compass width, place the compass at E and draw arcs that intersect the previous arcs.
    4. D I F Yes
  4. Label the intersection points above and below the line as F and G Then, draw a line through points F and G to form a right angle.
    5. Angle between line DF and DE is a 90-degree angle
    D I F Yes

Drawing a 45 degree angle

Constructing a 45 degree angle involves bisecting a 90 degree angle:

  1. Construct a 90 degree angle using the steps above.
  2. To bisect a 90 degree angle, place the compass at the vertex of the angle and draw an arc on both sides.
    3. H I J
  3. With the same radius, place the compass at the points where the arc crosses the arms and draw two arcs that intersect each other.
    4. H I J K
  4. Label the point of intersection as K Draw a line from the angle vertex to K to create a 45 degree angle.
    5. Angle between line HI and HK is a 45-degree angle
    H I J K

Constructing an angle of any given measure

Using a protractor is a simple way to draw an angle of any given measure, but it is necessary to know how to construct an angle using only a compass and a straight line, using special tools or methods such as angle bisectors or successive divisions.

Using successive divisions

You can construct angles of different measures by dividing known angles. For example, commonly used angles like 15°, 30°, etc., can be obtained by successive divisions of already constructed angles.

Drawing a 30 degree angle

To construct a 30 degree angle, you can start with a 60 degree angle and bisect it:

  1. Draw a 60 degree angle following the steps mentioned earlier.
  2. Using the same technique as for the 45 degree angle, obtain a 30 degree angle by bisecting the 60 degree angle.

By repeating this method, you can also create smaller angle measures, such as 15 degrees, by further bisecting the 30 degree angle.

Creation of real-life scenarios

Drawing angles isn't just limited to paper. It can also be applied to a variety of real-life situations:

  • Creating precise design plans in architecture and engineering.
  • Making angles in wood and metal work.
  • Planning a layout in graphic design.

Conclusion

Constructing angles using a compass and straightedge is fundamental in geometry, helping students and professionals to construct angles accurately and precisely. Understanding how to construct angles of 30, 45, 60, 90 degrees and more is a valuable skill that can be applied in a variety of fields, enhancing problem-solving abilities and spatial understanding.


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