Angles, Polygons and Circles


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Simple open curves made up of only two lines are called angles. Simple closed curves made up of more than two lines are called polygons. A circle is a simple closed curve formed by a point moving at the same distance from a fixed point.

Figures that can be drawn without lifting the pencil from the paper and without the use of a ruler are called curves. Curves that do not cross themselves are called 'simple curves'. In the following figure (i), (iii), (iv) and (vii) are simple curves, while (ii), (v) and (vi) are not simple curves. Also, (iii), (iv) and (vii) are examples of closed curves, while (i), (ii) (v) and (vi) are examples of open curves. 

An 'angle' is made up of two rays having a common end point. The two rays forming the angle are called 'sides' or 'arms' of the angle, and the common end point is called the 'vertex' of the angle. The portion that lies in between the two rays is called the interior of the angle. In naming an angle, the vertex is always written as the
middle letter.

Simple closed curves made up of only line segments are called polygons.Line segments that form the polygon are the sides of the polygon. Any two sides with a common end point are called the adjacent sides. The point of intersection of a pair of sides is called a vertex. The end points of the same side are adjacent vertices.The line segment joining any two non-adjacent vertices of a polygon is a diagonal.

A polygon made up of three line segments is called a triangle. The triangle in the diagram is ΔABC. Points A, B and C are the vertices.  AB _ , BC _ and CA _ are the sides, and ∠ABC, ∠BCA, ∠CAB are the angles of the triangle. A triangle has an interior and exterior.

A four-sided polygon is a quadrilateral. A quadrilateral is named in a cyclic manner. Points A, B, C and D are the vertices.  AB _ , BC _ , CD _ and DA _  are the sides, and ∠A, ∠B, ∠C and ∠D are the angles of quadrilateral ABCD.

A quadrilateral has, in all, four pairs of adjacent sides. They are AB and BC, BC and CD, CD and DA, and DA and AB. 

Also, it has two pairs of opposite sides. They are AB and DC, and BC and AD. Angles A and B, angles B and C, angles C and D, and angles D and A are said to be adjacent angles. Angles A and C, and angles B and D are said to be pairs of opposite angles.

A circle is formed by a point moving at the same distance from a fixed point. The fixed point is the centre of the circle. A circle is also a simple closed curve however, it does not have any sides or angles.

The line that forms the boundary of a circle is called its circumference. The part enclosed by the circumference of a circle is called the interior of the circle. The part left outside the circle is said to be the exterior of the circle. Some points may lie on the circumference of the circle. 

A line segment that joins the centre of the circle and a point on the circumference is called the radius of the circle. The radius of a circle is half of the diameter.

A chord is a line segment joining two points that lie on a circle.

A chord passing through the centre of the circle is called its diameter. A diameter is the longest chord of a circle.

An arc is a part of the circumference of a circle.

The part of a circle enclosed by two radii and an arc is called a sector. 

The part of a circle that is enclosed by a chord and an arc is called a segment of the circle.

A diameter of a circle divides it into two halves. Each half is called a semi-circle.

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