Fundamentals of Geometry

Main Points


geometry fundamentals


Point

A point is a fixed position. A dot represent a point. A is a point.

Line segment

The set of all points on a straight path between two end points is called a line segment. If the end points are A and B then the line segment is AB. A line segment has a definite length.

Line

When the end points of a line segment extends to infinity in both directions, the line segment is called a line. As infinity can not be reached hence a line has no end points. It has no definite length. It is represented by putting two arrows at the endpoints of a line segment as it can not be drawn on paper.

Ray

When only one endpoint of a line segment is extended to infinity it is called a ray. It is represented by putting one arrow at one of the endpoints of a line segment as it can not be drawn on paper. It has no definite length.

Angle and its types


Angles and its types


Angle

Two rays OA and OB starting from a common point describe an angle between them. The common point is O. OA and OB are called the arms of the angle and O is called the vertex.
  • Right angle

    An angle whose measure is 90° is called a right angle.
  • Acute angle

    An angle whose measure is more than 0° and less than 90° is called an acute angle.
  • Obtuse angle

    An angle whose measure is more than 90° and less than 180° is called an obtuse angle.
  • Straight angle

    An angle whose measure is 180° is called a straight angle.
  • Reflex angle

    An angle whose measure is more than 180° and less than 360° is called a reflex angle.
  • Complete angle

    An angle whose measure is 360° is called a complete angle.



adjacent angles

Adjacent angles

Adjacent angles have a common arm and the two angles are on the opposite sides of the common arm.


Vertically opposite angles

Vertically Opposite angles

If two lines AC and BD intersect at the point O then the angles AOB and DOC form one pair of vertically opposite angles. Similarly angles AOD and BOC form another pairs of vertically opposite angles.
The vertically opposite angles are equal.
∠AOB = ∠DOC and ∠AOD = ∠BOC

Complementary angles

Two angles are said to be complementary if the sum of their angles is 90°

Supplementary angles

Two angles are said to be supplementary if the sum of their angles is 180°
linear pairs

Linear Pair

If the sum of two adjacent angles is equal to 180°, they are said to form a linear pair.
∠AOB + ∠BOC = 180° as AOC is a straight line.


angles around a point

Angles about a point

The sum of all angles formed about a point is equal to two right angles.
∠1 + ∠2 + ∠3 + ∠4 = 360°

Parallel Lines and Traversal


When two lines lie in the same plane and don't intersect each other even after producing to infinity are called parallel lines.

Traversal

Traversal

A straight line that intersects two or more straight lines is called a traversal. Let AB and CD be two parallel lines and EF be the traversal. Then, we find the following angles in it
  • Pairs of corresponding angles

    (∠1,∠5), (∠2,∠6),(∠4,∠8) and (∠3,∠7).
    Corresponding angles are equal.

    ∠1 = ∠5, ∠2 = ∠6, ∠4 = ∠8 and ∠3 = ∠7
  • Pairs of alternate interior angles

    (∠3,∠5) and (∠4,∠6).
    Alternate interior angles are equal.

    ∠3 = ∠5 and ∠4 = ∠6
  • Pairs of alternate exterior angles

    (∠2,∠8) and (∠1,∠7)
    Alternate exterior angles are equal.

    ∠2 = ∠8 and ∠1 = ∠7
  • Pairs of consecutive interior angles

    (∠4,∠5) and (∠3,∠6).
    Co-interior angles are supplementary.

    ∠4+∠5 = 180° and ∠3+∠6 = 180°



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