Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Geometry → Constructions ↓


Constructing Angles


Geometry is a wonderful world of points, lines, shapes, and angles. Learning geometry in school helps us understand the world better. One of the fundamental skills in geometry is drawing angles using basic tools like a ruler and compass. In this lesson, we are going to explore how we can draw angles. Let's dive into this step-by-step process and understand it in a very simple way.

Understanding angles

Before drawing an angle, let us understand what an angle is. An angle is formed when two lines meet at a common end point called the vertex. The two straight lines are called the sides of the angle. Angles are measured in degrees. There are different types of angles based on the degree measure:

  • Acute angle: An angle whose measure is more than 0 degrees and less than 90 degrees.
  • Right angle: An angle that is exactly 90 degrees.
  • Obtuse angle: An angle that is more than 90 degrees but less than 180 degrees.
  • Straight angle: An angle that is exactly 180 degrees.

Tools required for angling

To draw an angle you will need a few basic tools. These are:

  1. Compass: An instrument used for drawing circles and arcs.
  2. Ruler: A straight tool for drawing straight lines.
  3. Pencil: For marking points and drawing lines.

Basic steps for constructing different angles

Drawing a 60 degree angle

Let's start by drawing a 60 degree angle. You usually start with a line segment. Follow these steps:

  1. Draw a line segment: Start by drawing a straight line segment, say AB. This will be one side of the angle.
  2. Set the compass width: Open the compass to the appropriate width. This width can be arbitrary but should be convenient for drawing the arc.
  3. Draw an arc: With the compass point at A, draw an arc intersecting AB.
  4. Draw a second arc from the first arc: Without changing the width of the compass, place the compass point at the intersection of the arc with line AB and draw a second arc that intersects the first arc.
  5. Mark the intersection: The point where the second arc intersects the first arc is point C.
  6. Draw the second side: Connect point A to point C. Angle BAC is a 60 degree angle.

Drawing a 90 degree angle

Constructing a 90 degree angle, also called a right angle, involves the following steps:

  1. Draw the base line segment: Suppose we draw a line segment PQ.
  2. Draw an arc: Place the compass point at P and draw an arc that intersects PQ at a point, designate this point R.
  3. Draw a second arc from the intersection: Without changing the compass width, place the compass at R and draw a second arc, labeling the intersection with the previous arc as S.
  4. Repeat again: Without changing the width of the compass, place the point at S and draw another arc to intersect the arc drawn from R. Label it T.
  5. Draw a perpendicular line: Place the compass point at T and draw an arc that intersects the outer arc drawn from R at a new point U.
  6. Connect the points: Connect the vertices P and U. The angle TPU is exactly 90 degrees.

Drawing a 30 degree angle

To construct a 30 degree angle, we work with what we know about a 60 degree angle:

  1. Start with a 60 degree angle: Follow the same steps as for drawing a 60 degree angle.
  2. Construct the external bisector: Set the compass to an arbitrary width, draw arcs from each open angle side that intersect. The point at which these arcs intersect is used to draw a line back to the vertex, which divides the angle in half. This means you have constructed two angles of 30 degrees.

Drawing a 45 degree angle

A 45 degree angle is half of a 90 degree angle. A 90 degree angle is divided like this:

  1. Make a 90 degree angle: As instructed previously.
  2. Bisection process: Similar to the 30-degree construction, use a compass to bisect a 90-degree angle. Draw arcs from the two sides of the angle and mark their intersection.
  3. Create a bisection line: Connect the original vertex and the point where the arcs intersect. This will create a 45 degree angle.

Using postulates and theorems for angles

Postulates and theorems in geometry allow us to construct and verify angles. Here are some important ones:

Angle bisector theorem

The angle bisector theorem tells us that if you have an angle and you bisect it, then any point on the bisector line is equidistant from both sides of the angle.

Corresponding angles postulate

This means that when two parallel lines are cut by a transversal, the angles in the same relative position are equal.

Some common problems and solutions

Constructing an angle without a protractor

When you need to construct an angle but don't have a protractor, using a compass and straightedge is a reliable method. As demonstrated, many angles can be constructed by first constructing vertical (90-degree) or equilateral (60-degree) angles and then bisecting them.

Common mistakes

  • Incorrect compass width: Ensuring that the compass width remains unchanged during construction is critical to accuracy.
  • Drawing uneven arcs: It is important to keep the arcs neat and uniform in size when drawing for more accurate intersections.

Benefits of angling

Constructing angles helps develop geometric understanding and solve geometry problems.

  • Lays a strong foundation for understanding the properties of geometric shapes.
  • Helps in accurate depiction of architectural designs, engineering plans and crafts.
  • Facilitates advanced understanding of optometry, surveying and molecular chemistry where angle understanding is critical.

Conclusion

The process of constructing angles using simple tools such as a compass and ruler is essential in geometry and serves as the basis for many advanced math and science-related fields. By understanding and practicing constructing angles, students not only gain academic skills but also equip themselves with practical abilities to creatively solve everyday problems. Whether constructing a simple 60-degree angle or dividing angles into equal parts, the skills learned here form an essential link between arithmetic and spatial reasoning.

Practice problems

  • Construct an angle of 75 degrees without using a protractor.
  • Bisect the given angle of 120 degrees and verify through construction.
  • Draw two parallel lines and a transversal. Measure alternate interior angles and verify their equality by construction.
  • Draw a triangle in which each interior angle is 60 degrees, using only a compass and a straight line.

Grade 8 → 3.2.3


U
username
0%
completed in Grade 8

← Prev (3.2.2)
Constructing Bisectors
Next (3.2.4) →
Constructing Triangles

Comments 



Constructing Angles

https://www.buddymath.com/g8/3.2.3?bmal=en