Angles, Polygons and Circles

Objective

According to historians, the geometrical ideas shaped up in ancient times, probably due to the need in art, architecture and measurement. Construction of magnificent palaces, temples, lakes, dams and cities, art and architecture propped up these ideas. Now let us study few Geometrical Concepts, they are

  1. Polygon
  2. Angle
  3. Triangle
  4. Quadrilateral
  5. Circle

Polygon

A polygon is a simple closed curve made up entirely of line segments. A polygon has at least 3 sides. The line segments forming a polygon are called its sides and the points at which the sides meet are called vertices.

In the example the sides of polygon ABCDE are AB, BC, CD, DE and EA. Sides AE  and ED meet at E, so E is a vertex of the polygon ABCDE. Any two sides with a common end point are called the adjacent sides of the polygon.

The pairs of adjacent sides are AB and BC; BC and CD; CD and DE; DE and EA; EA  and AB. The end points of the same side of a polygon are called the adjacent vertices. The pairs of adjacent vertices are: A and B; B and C; C and D; D and E; E and A. The joins of pairs of vertices which are not adjacent are called the diagonals  of the polygon. The diagonals are
 

Angles 

Angles are made when corners are formed. When two rays have a common end point, then two rays together are said to form an angle. An angle is made up of two rays starting from a common end point. The two rays forming the angle are called the arms or sides of the angle. The common end point is the vertex of the angle. To show an angle we use a small curve at the vertex. Symbol for angle is " ".

 

To name an angle two points, one on each side and the vertex are used. Thus, ∠ POQ  is a better way of naming the angle instead of ∠ O. In specifying the angle, the vertex is always written as the middle letter. The angle also has three parts associated with it, which are the interior, exterior and on the angle.

Triangle

A triangle is a three-sided polygon. In fact, it is the polygon with the least number of sides. Symbol for triangle is.

A triangle has three sides, three angles and three vertices. Being a polygon, a triangle has an exterior and an interior. 

The interior angles are those on the inside of the triangle. The interior angles of a triangle always add up to 180°

 

 In triangle ABC,

  •  The three sides of the triangle are 
  • The three angles are ∠BAC, ∠BCA and ∠ABC.
  • The points A, B and C are called the vertices of the triangle.

Quadrilateral

A four sided polygon is a quadrilateral. It has 4 sides, 4 vertices and 4 angles. Structures like electric towers make use of triangular shapes and not quadrilaterals as the triangle is very rigid and hence a strong shape. When we push inward at any one vertex of the triangle the triangle does not get distorted.

The vertices are named in a cyclic manner.

  • Quadrilateral ABCD has four sides.
  •  It has four angles ∠A, ∠B, ∠C and ∠D.
  • In any quadrilateral ABCD,
    • Adjacent sides are.
    • Opposite sides are AB and DC; and BC and AD. 
    • Opposite angles are ∠A and ∠C; and ∠B and ∠D. 
    • Adjacent angles are ∠A and ∠B; ∠B and ∠C; ∠C and ∠D; ∠A. and ∠D;∠D and ∠A.

Circles

A circle is a simple closed curve which is not a polygon. Every point on the circle is at equal distance from the centre. The radius is a line segment joining the centre to a point on the circle. The plural of ‘radius’ is radii. A line segment which connects two points on a circle and passes through the centre is called the Diameter. Diameter is double the size of a radius. The chord is a line segment connecting two points on a circle.  All diameters are chords. The diameter is the longest chord of the circle.

 

An arc is a portion of circle. A circle has three parts - the interior, exterior and on the circle. A region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides is called a sector. A region in the interior of a circle enclosed by a chord and an arc is called a segment of the circle. The distance around a circle is its circumference. A diameter of a circle divides it into two equal parts called  semi-circle. A semi-circle is half of a circle, with the end points of diameter as part of the boundary. 

Summary 

  1. A polygon is a simple closed curve made up of line segments. Here,
    • The line segments are the sides of the polygon.
    • Any two sides with a common end point are adjacent sides.
    • The meeting point of a pair of sides is called a vertex.
    • The end points of the same side are adjacent vertices.
    • The joint of any two non-adjacent vertices is a diagonal.
  2. An angle is made up of two rays starting from a common end point.
  3. A triangle is a three-sided polygon.
  4. A quadrilateral is a four-sided polygon.
  5. A circle is the path of a point moving at the same distance from a fixed point.
    • The fixed point is the centre, the fixed distance is the radius and the distance around the circle is the circumference.
    • A chord of a circle is a line segment joining any two points on the circle.
    • A diameter is a chord passing through the centre of the circle.
    • A sector is the region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides.
    • A segment of a circle is a region in the interior of the circle enclosed by an arc and a chord.
    • The diameter of a circle divides it into two semi-circles.

 

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