Basic Constructions - Angle Bisector, Perpendicular Bisector, Angle of 60° Last Updated : 31 Aug, 2021 Comments Improve Suggest changes Like Article Like Report Most of the time we use diagrams while depicting the shapes and scenarios in mathematics. But they are not precise, they are just a representation of the actual shape without proper measurements. But when we are building something like a wooden table, or map of a building is to be constructed. It needs to be precise in measurements. For such cases, we need to learn to do some basic constructions. Let's study them in detail. Introduction to Constructions To draw polygons, angles, or circles some basic instruments are needed which are given in our geometry box. A geometry box should contain the following instruments: 1. A ruler or graduated scale: It should have centimetres and millimetres marked off on one side and the other side should have inches and their parts marked off on it. 2. Pair of Set Squares: One should have 60°, 90° and 30° as its angles and the other one should have 90°, 45° and 45°. 3. Divider Dividers have a jointed pair of legs, each leg has a sharp point. They can be used for scribing a circles and taking off and transferring the dimensions. 4. Compass Compass is used for inscribing circles or arcs. As similar to dividers, they can also be used for taking off and transferring the dimensions. 5. Protractor This instruments helps us in measuring the angles. It has degrees marked from 0 to 180 and from 180 to 0. It has readings from both left to right and right to left. Both the readings supplement each other. For learning about the basic constructions we will mostly use three of them. Graduated Scale, Compass, and a Protractor. Let's see some basic constructions. Basic Constructions We know that a bisector is a line that divides anything into two equal parts, be it an angle or a line segment. Let's see how to make bisectors of any given angle using our instruments from the geometry box. Construction 1: Angle Bisector Let's say we are given an angle PQR and the goal is to construct a bisector of the given angle. Steps for Construction: Step 1. Take Q as centre and draw an arc that intersects the rays QP and QR with any radius. Let's name the intersection points T and S. Step 2. Now take T and S as the Centre and with the radius that is more than half of the length of TS. Draw arcs such that they intersect, let's call that intersection U. Step 3. Now join QU, this is our required bisector. Construction 2: Perpendicular bisector of the given line. A perpendicular bisector is a line that bisects the given line and is perpendicular to it. Given a line PQ, the goal is to construct a perpendicular bisector. Steps for Construction: Step 1. Let's take P and Q as the centre and take any radius that more than half the length of PQ. Now draw arcs on both sides of the line PQ and let them intersect at A and B respectively. Step 2. Now Join AB and let it intersect at M on PQ. This is our required perpendicular bisector. Construction 3: An angle of 60° The goal in this construction is to construct an angle of 60° from a given ray PQ. Steps of Construction: Step 1. With P as the centre and some arbitrary radius, construct a circular arc that intersects PQ at S. Step 2. Now take S as the centre and the same radius as above. Draw an arc on the arc already drawn. Let's say the point of intersection of both arcs is T. Step 3. Draw a ray PR passing through T. This will give us the required angle of 60°. Let's see some examples on these concepts. Sample Problems Question 1: In the figure below, ∠PQR is divided into many parts. Determine the bisector for ∠PQR. Solution: We know that bisector of an angle divides it into two equal parts. Notice that, ∠PQT = ∠SQT + ∠PQS ∠PQT = 30° + 20° ∠PQT = 50° and, ∠TQR = ∠TQU + ∠UQR ∠TQR = 35° + 15° ∠TQR = 50° This the ray QT is the bisector of the angle PQR. Question 2: State True or False for the statements below. Perpendicular bisector bisects a line segment. An angle bisector divides an angle into two equal parts. Solution: Statement 1: False According to the definition of perpendicular bisector, it is a line that divides a line segment into two equal parts and is also perpendicular to it. Statement 2: True An angle bisector divides an angle into two equal parts. Question 3: Construct an angle of 30° using the construction techniques mentioned above. Solution: Steps of Construction: Step 1. Using the construction technique mentioned above for creating 60° angle. Let's call that angle ∠CAB. Step 2. Now, since the angle is 60° and out goal is to construct an angle of 30°. We need to bisect this angle using the technique mentioned above. Now the ray AE gives us the angle of 30°. Question 4: Construct an angle of 135°. Solution: An angle of 135° can be made using one 90° and one 45°. Let's make a straight line PQ with a point R in between. Now we have made two 90° angles. We need to bisect on the of angles to make a 45°. Thus, ∠PRE gives us the required angle 135°. Comment More infoAdvertise with us Next Article Basic Constructions - Angle Bisector, Perpendicular Bisector, Angle of 60° A anjalishukla1859 Follow Improve Article Tags : Mathematics School Learning Class 9 Maths-Class-9 Similar Reads CBSE Class 9 Maths Revision Notes CBSE Class 9th Maths Revision Notes is an important phase of student’s life when they’re at a turning point in their life. The reason being class 9 is the foundation level to succeed in class 10. As you know, students must complete Class 9 in order to sit for Class 10 board examinations. 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Egyptian and Indian civilizations were more focused on using geometry as a tool. Euclid came and changed the way people used to think in geometry. Instead of making it 6 min read Chapter 6: Lines and AnglesLines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang 9 min read Types of AnglesTypes of Angles: An angle is a geometric figure formed by two rays meeting at a common endpoint. It is measured in degrees or radians. It deals with the relationship of points, lines, angles, and shapes in space. Understanding different types of angles is crucial for solving theoretical problems in 10 min read Pairs of Angles - Lines & AnglesWhen two lines share a common endpoint, called Vertex then an angle is formed between these two lines and when these angles appear in groups of two to display a specific geometrical property then they are called pairs of angles. Understanding these angle pairs helps in solving geometry problems invo 8 min read Transversal LinesTransversal Lines in geometry is defined as a line that intersects two lines at distinct points in a plane. The transversal line intersecting a pair of parallel lines is responsible for the formation of various types of angles that, include alternate interior angles, corresponding angles, and others 7 min read Angle Sum Property of a TriangleAngle Sum Property of a Triangle is the special property of a triangle that is used to find the value of an unknown angle in the triangle. It is the most widely used property of a triangle and according to this property, "Sum of All the Angles of a Triangle is equal to 180º." Angle Sum Property of a 8 min read Chapter 7: Triangles Triangles in GeometryA triangle is a polygon with three sides (edges), three vertices (corners), and three angles. It is the simplest polygon in geometry, and the sum of its interior angles is always 180°. A triangle is formed by three line segments (edges) that intersect at three vertices, creating a two-dimensional re 13 min read Congruence of Triangles |SSS, SAS, ASA, and RHS RulesCongruence of triangles is a concept in geometry which is used to compare different shapes. It is the condition between two triangles in which all three corresponding sides and corresponding angles are equal. Two triangles are said to be congruent if and only if they can be overlapped with each othe 9 min read Theorem - Angle opposite to equal sides of an isosceles triangle are equal | Class 9 MathsIn geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Exampl 4 min read Triangle Inequality Theorem, Proof & ApplicationsTriangle Inequality Theorem is the relation between the sides and angles of triangles which helps us understand the properties and solutions related to triangles. Triangles are the most fundamental geometric shape as we can't make any closed shape with two or one side. Triangles consist of three sid 8 min read Chapter 8: QuadrilateralAngle Sum Property of a QuadrilateralAngle Sum Property of a Quadrilateral: Quadrilaterals are encountered everywhere in life, every square rectangle, any shape with four sides is a quadrilateral. We know, three non-collinear points make a triangle. Similarly, four non-collinear points take up a shape that is called a quadrilateral. It 9 min read QuadrilateralsQuadrilateral is a two-dimensional figure characterized by having four sides, four vertices, and four angles. It can be broadly classified into two categories: concave and convex. Within the convex category, there are several specific types of quadrilaterals, including trapezoids, parallelograms, re 12 min read Parallelogram | Properties, Formulas, Types, and TheoremA parallelogram is a two-dimensional geometrical shape whose opposite sides are equal in length and are parallel. The opposite angles of a parallelogram are equal in measure and the Sum of adjacent angles of a parallelogram is equal to 180 degrees.A parallelogram is a four-sided polygon (quadrilater 10 min read Rhombus: Definition, Properties, Formula and ExamplesA rhombus is a type of quadrilateral with the following additional properties. All four sides are of equal length and opposite sides parallel. The opposite angles are equal, and the diagonals bisect each other at right angles. A rhombus is a special case of a parallelogram, and if all its angles are 6 min read Trapezium: Types | Formulas |Properties & ExamplesA Trapezium or Trapezoid is a quadrilateral (shape with 4 sides) with exactly one pair of opposite sides parallel to each other. The term "trapezium" comes from the Greek word "trapeze," meaning "table." It is a two-dimensional shape with four sides and four vertices.In the figure below, a and b are 8 min read Square in Maths - Area, Perimeter, Examples & ApplicationsA square is a type of quadrilateral where all four sides are of equal length and each interior angle measures 90°. It has two pairs of parallel sides, with opposite sides being parallel. The diagonals of a square are equal in length and bisect each other at right angles.Squares are used in various f 5 min read Kite - QuadrilateralsA Kite is a special type of quadrilateral that is easily recognizable by its unique shape, resembling the traditional toy flown on a string. In geometry, a kite has two pairs of adjacent sides that are of equal length. This distinctive feature sets it apart from other quadrilaterals like squares, re 8 min read Properties of ParallelogramsProperties of Parallelograms: Parallelogram is a quadrilateral in which opposite sides are parallel and congruent and the opposite angles are equal. A parallelogram is formed by the intersection of two pairs of parallel lines. In this article, we will learn about the properties of parallelograms, in 9 min read Mid Point TheoremThe Midpoint Theorem is a fundamental concept in geometry that simplifies solving problems involving triangles. It establishes a relationship between the midpoints of two sides of a triangle and the third side. This theorem is especially useful in coordinate geometry and in proving other mathematica 6 min read Chapter 9: Areas of Parallelograms and TrianglesArea of a Triangle | Formula and ExamplesThe area of the triangle is a basic geometric concept that calculates the measure of the space enclosed by the three sides of the triangle. The formulas to find the area of a triangle include the base-height formula, Heron's formula, and trigonometric methods.The area of triangle is generally calcul 6 min read Area of Parallelogram | Definition, Formulas & ExamplesA parallelogram is a four-sided polygon (quadrilateral) where opposite sides are parallel and equal in length. In a parallelogram, the opposite angles are also equal, and the diagonals bisect each other (they cut each other into two equal parts).The area of a Parallelogram is the space or the region 8 min read Figures on the Same Base and between the Same ParallelsA triangle is a three-sided polygon and a parallelogram is a four-sided polygon or simply a quadrilateral that has parallel opposite sides. We encounter these two polynomials almost everywhere in our everyday lives. For example: Let's say a farmer has a piece of land that is in the shape of a parall 6 min read Like