Geometry

Category : 6th Class

                                                                                   Geometry

 

Learning Objectives

  • Geometry
  • Types of angles
  • Parts of Circle

 

Geometry

The branch of mathematics which deals with mathematical objects like points, lines, panes and space is called geometry.

Point: A point is to be thought of as a location in space. In other words, a point determines location in a space.

Line segment: Let A and B be two point on a plane. Then the straight path between points a and B is known as Sine segment AB.

The above line segment is denoted as\[\overline{AB}\].

 

Ray: A line segment extended endlessly in one direction Is. called a ray.

The above ray Is denoted as \[\overrightarrow{AB}\].

 

Line: A line is a straight path that extends on and on in both directions endlessly.

The above ray Is denoted as \[\overrightarrow{AB}\].

 

Line: A line is a straight path that extends on and on in both directions endlessly.

 

Parallel lines: If two or more lines do not meet each other however far they are extended, then they are called parallel lines

Intersecting lines: If two or more lines meet each other at one point they are called intersecting lines.

Concurrent lines: Three or more lines in a place are concurrent if all of them pass through the same point. The common point is called point of concurrence.

Collinear points: Three or more points which lie on the same line are collinear, and the line is called the line of collinearity for the given points.

Angle: The figure formed by two rays with the same initial point is called an angle.

Types of angles

Acute angle: An angle which measures more than \[=30{}^\circ \times 3=90{}^\circ .\]but less than \[90{}^\circ \]is called an acute angle.

Right angle: An angle which measure \[90{}^\circ \] is called a right angle.

Obtuse angle: An angle which measures more than \[90{}^\circ \] but less than \[180{}^\circ \] is called an obtuse angle.

Straight angle: An angle which measures \[180{}^\circ \] is called a straight angle.

Reflex angle: An angle which measures more than \[180{}^\circ \] but less than \[360{}^\circ \] is called a reflex angle.  
 

Complete angle: An angle which measures \[360{}^\circ \] is called a complete angle.

Zero angle: An angle which measures \[0{}^\circ \]is called zero angle.
 

Triangle: A triangle is a closed plan figure bound by three lines segments. In fact triangle is a polygon.  

Types of trianlges

Equilateral triangle: A triangle whose all sides are of equal length is known as equilateral triangle.

Isosceles triangle: A triangle whose all sides are of equal length is known as isosceles triangle.

Scalene triangle: A triangle whose all sides are of different length is known as scalene triangle.

Acute angled triangle: If all the three angles of a triangle are acute then the triangle is known as acute angled triangle.

Obtuse angled triangle: The triangle having an obtuse angle in known as obtuse angled triangle. 

Right angled triangle: The triangle having a right angle is known as right angled triangle.   

Quadrilateral: A quadrilateral is a closed plane figure bounded by four lines segments or a quadrilateral is a four sided polygon.   

Types of quadrilatreral

Trapezium: A quadrilateral, in which one and only one pair of opposite sides are parallel, is known as trapezium.  

Parallelogram: A quadrilateral in which both pair of opposite sides are parallel is called a parallelogram.

Rhombus: A parallelogram, whose all sides are of equal length, is known as rhombus.  

Rectangle: A parallelogram, in which each of the angles is a right angle, is known as rectangle.  

Square: A parallelogram, in which all sides are equal and each angle is a right angle, is known as square.

Circle: A circle is simple closed curve all of whose points are at the same distance from a given point in the same plane.

                       

Parts of a circle

Centre: The fixed point in the plane which is equidistant from every point lying on the boundary of the circle is called centre of the circle.

 

Radius: The line segment joining the centre of the circle to any point on the circle is the radius of that circle.

Chord: A line segment joining any two points on a circle is called chord of the circle.

Diameter: A chord which passes through the centre of a circle is called a diameter of that circle.

Secant: A line which intersects the circles at two distinct points is called a secant.

Arc: An arc is a part of a circle.

Segment: A region in the interior of the circle enclosed by a chord and an arc is called a segment of the circle.

Sector: A region in the interior of the circle enclosed by two radii and an arc is called sector of the circle.

Semicircle: A diameter divides a circle into two equal parts. Each part is called a semicircle.

Circumference: The length of the boundary of the interior of a circle is called circumference of the circle.

Concentric circles: Two or more circles with the same centre are called concentric circles.

 

 

Three Dimensional Shapes

                       

                                   

Cuboid: Cuboid is a three dimensional solid shape. A rectangular wooden box is an example of a cuboid. A cuboid has 6 rectangular faces, 12 edges and 8 vertices.

 

Cube: A cuboid whose length, breadth and height are equal is called a cube, A cube has 5 square faces, 12 edges and 8 vertices.

 

Cylinder: A cylinder is three dimensional geometric figure that has two congruent and parallel bases. It has no vertex, two circular faces, one curved face and two curved edges.

 

Sphere: An object which is in the shape of a ball is said to have the shape of sphere. It has a curved surface but no vertex and no edge.

 

Cone: A cone is a three dimensional geometric shape that tapers smoothly from a fiat to a point called the apex or vertex. It has one vertex, one curved edge, one curved face and one flat face.

Pyramid: A pyramid is a solid whose base is a plane rectilinear figure and whose side faces are triangles having a common vertex called the vertex of the pyramid.

 

Symmetry

A figure drawn on the paper is symmetric, if it can be folded in such a way that the two halves of the figure exactly cover each other. The line of fold is called axis of the symmetry,

 

Symmetry of some geometrical shapes

 

Line segment: A line segment is symmetrical about its perpendicular bisector.

 

Angle: An angle with equal arms is symmetrical about the bisector of the angle,

           

Scalene triangle: A scalene triangle has no line of symmetry.

Parallelogram: A parallelogram has no line of symmetry.

 

Table of Symmetry of Geometrical Shapes

 

Shape

Figure

Number of lines of symmetry

Equilateral triangle

3

Isosceles triangle

1

Square

4

Rectangle

2

Rhombus

2

Semicircle

1

Circle

Infinite

 

Commonly Asked Questions

 

Which one of the following is the complementary angle of \[\mathbf{30}{}^\circ \] ?

(a) \[30{}^\circ \]          (b) \[60{}^\circ \]

(c) \[90{}^\circ \]           (d) All of these

(e) None of these

 

Answer (b)

Explanation: Complementary angle of \[30{}^\circ =90{}^\circ -30{}^\circ =60{}^\circ .\]

 

In the options given below the pair of angles are given. Find the complementary pair.

(a)\[51{}^\circ ,25{}^\circ \]                

(b) \[61{}^\circ ,29{}^\circ \]

(c) \[45{}^\circ ,55{}^\circ \]                

(d) All of these

(d) None of these

 

Answer: (b)

Explanation: If the sum of pair of angles is \[90{}^\circ \] then the pair is called cornplementary pair of angles. The sum of angles \[61{}^\circ +29{}^\circ =90{}^\circ .\] Hence the pair of angles \[61{}^\circ ,29{}^\circ \] is a complementary pair of angles,

 

Choose the pair of supplementary angle from the options given below?

(a) \[100{}^\circ ,200{}^\circ \]            

(b) \[105{}^\circ ,75{}^\circ \]

(c) \[30{}^\circ ,180{}^\circ \]              

(d) All of these

(e) None of these

 

Answer: (b)

Explanation: The sum of angles of the pair \[105{}^\circ ,75{}^\circ =105{}^\circ +75{}^\circ =180{}^\circ \] Hence, the pair of angles \[105{}^\circ ,75{}^\circ \] is a supplementary pair of angles.

 

Find the measurement of A if B and C are given when A, B and C are the angle of the triangle.

(a)\[A=180{}^\circ -\left( B+C \right)\]    (b) \[A=180{}^\circ +\left( B-C \right)\]

(c) \[A=180{}^\circ +C-B\]       

(d) All of these

(e) None of these

Answer (a)

Explanation: The sum of angles of the triangle \[=180{}^\circ \]. Hence, the angle \[A=180{}^\circ -\left( B+C \right).\]       

 

If the base of a triangle is 18 cm and distance between base and opposite vertex is 10 cm, what will be the area of triangle?

(a) \[206\text{ }c{{m}^{2}}\]                              

(b) \[10\text{ }c{{m}^{2}}\]

(c) \[90\text{ }c{{m}^{2}}\]                                  

(d) All of these

None of these

Answer (c)

Explanation: The distance between base and opposite vertex is height,

\[The\,area={{\frac{1}{2}}^{~}}\times base\times height\]

\[=\frac{1}{2}18\times 10=90c{{m}^{2}}.\]

Other Topics

Notes - Geometry
  15 10



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