Fundamentals of geometry

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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

 

Important definitions

Point

A point is a zero-dimensional object which does not have length, width, area or volume which typify objects of higher dimension. It just specifies an exact location in the space.

 

Line

A line is a straight curve which extends indefinitely in both directions. The concept of a line is an extension of the concept of points. A line is a collection of points that extends without limit in a straight formation. A line that is bounded by one end point and indefinitely extends on the other side is called a ray.

 

Line segment

A line segment is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Unlike a line, whose length is infinite, a line segment has finite length. The terms line and line segment are typically used interchangeably. The point that is equidistant from both each endpoint is the midpoint of the line segment. Because a midpoint splits the line segment into two equal halves, the midpoint is said to bisect the line segment.

 

Angle

An angle is a figure formed by two rays sharing a common point. This point is called the vertex of the angle. The word angle can also indicate the angular magnitude of the above mentioned geometric configuration and is denoted by a number.
 

 

Types of angles

  • Acute angle -An angle less than 90 is called an acute angle.
  • Obtuse angle-An angle greater than 90 but less than 180 is called an obtuse angle.
  • Right angle-An angle equal to 90 is called a right angle.
  • Reflex angle-An angle greater than 180 but less than 360 is called a reflex angle.
  • Complementary angles-Two angles whose sum is 90 are called complementary angles. In the given figure,
  • Supplementary angles-Two angles whose sum is 180 are called supplementary angles.

Properties of angles

  1. The sum of angles made by any number of lines on one side of a straight line subtending at a point will always be equal to 180180∘.
  2. The sum of angles formed by all straight lines meeting at a point will always be equal to 360 .
  3. When two lines or line segments intersect, two pairs of congruent equal angles are formed. The angles in each pair of the congruent angles formed by the intersection of two lines are called vertical angles. Vertical angles are equal to each other.

 

Perpendicular bisector

When two lines intersect at right angles, then they are said to be perpendicular. When one line divides another line into two equal halves, then the first line is called the bisector of the second line. When the bisector is perpendicular, it is called the perpendicular bisector. Angles subtended by the end points of the second line on any point on the perpendicular bisector are always equal.

 

Angular bisector

When a straight line drawn at the vertex of a given angle, divides the angle into two equal halves, then the line is said to be the angular bisector of the given angle. Any point on the angular bisector is equidistant from the arms of the angle.

 

Parallel lines

Two lines in the same plane are said to be parallel if they never intersect each other. When a straight line cuts two or more parallel lines, the line is said to be a transversal. We can define the relationships between various angles formed between the parallel lines and transversal as follows

Alternate exterior angles are pairs of congruent angles on opposite sides of a transversal, outside of the space between the parallel lines. In the figure above, there are two pairs of alternate exterior angles:Angles 1 and 8, angles 2 and 7.

Alternate interior angles (Z type angles) are pairs of congruent angles on opposite sides of the transversal in the region between the parallel lines. In the figure above, there are two pairs of alternate interior angles:Angles 3 and 6, 4 and 5.

Corresponding angles are congruent angles on the same side of the transversal. In the figure above, there are four pairs of corresponding angles:Angles 1 and 5, 2 and 6, 3 and 7 and 3 and 8.


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