Parallel Lines and a Transversal

Parallel lines with transversal

• A point is that which has no part. 
• A line is a breadthless length. The ends of a line are points. 
• A straight line is a line which lies evenly with the points on itself. 
• A surface is that which has length and breadth only. 
• The edges of a surface are lines. 
• A plane surface is a surface which lies evenly with the straight lines on itself. 

Though Euclid defined a point, a line and a plane, but the definitions are not accepted by mathematicians. Therefore these terms are taken as undefined. 

An axiom is a statement accepted as true without proof, throughout mathematics. 

A postulate is a statement accepted as true without proof, specifically in geometry. 

Euclid’s Axioms: 

• Things which are equal to the same things are equal to one another. 
• If equals are added to equals, the wholes are equal. iii. If equals are subtracted from equals, the remainders are equal. iv. Things which coincide with one another are equal to one another. 
• The whole is greater than a part. 
• Things which are double of same things are equal to one another. 
• Things which are halves of same things are equal to one another. 

Parallel lines with a transversal

• ∠1=∠5,∠2=∠6,∠4=∠8 and ∠3=∠7(Corresponding angles)
• ∠3=∠5,∠4=∠6 (Alternate interior angles)
• ∠1=∠7,∠2=∠8 (Alternate exterior angles)

Lines parallel to the same line

– Lines that are parallel to the same line are also parallel to each other.

Introduction to Geometry

Angles and types of angles

When 2 rays originate from the same point at different directions, they form an angle.

• The rays are called arms and the common point is called vertex
• Types of angles :
  1. Acute angle 0∘<a<90∘
  2. Right angle a=90
  3. Obtuse angle : 90∘<a<180∘
  4. Straight angle =180∘
  5. Reflex Angle 180∘<a<360∘
  6. Angles that add up to 90∘ are complementary angles
  7. Angles that add up to 180∘ are called supplementary angles.

Intersecting Lines and Associated Angles

Intersecting and Non-Intersecting lines

When 2 lines meet at a point they are called intersecting

When 2 lines never meet at a point, they are called non-intersecting or parallel lines

Adjacent angles

2 angles are adjacent if they have the same vertex and one common point.

Adjacent angles

Linear Pair

When 2 adjacent angles are supplementary, i.e they form a straight line (add up to 180), they are called a linear pair.

Vertically opposite angles

When two lines intersect at a point, they form equal angles that are vertically opposite to each other.

Basic Properties of a Triangle

Triangle and sum of its internal angles

Sum of all angles of a triangle add up to 180

Exterior angle of a triangle = sum of opposite internal angles

If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles

∠4 is the exterior angle ∠4=∠1+∠2

Theorems are statements which are proved using definitions, axioms, previously proved statement and deductive reasoning. 

Euclid’s 5 Postulates: 

A straight line may be drawn from any one point to any other point.  

A terminated line can be produced indefinitely. 

A circle can be drawn with any centre and any radius. iv. All right angles are equal to one another. 

v. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.  This is known as the parallel postulate.  

Given two distinct points, there is a unique line that passes through them. 

Two distinct lines cannot have more than one point in common. 

Equivalent version of Euclid’s fifth postulate: 

For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l. 

This result is also known as ‘Playfair’s Axiom‘. 

Two distinct intersecting lines cannot be parallel to the same line.  

All attempts to prove Euclid’s fifth postulate using first four postulates failed and led to several other geometries called non Euclidean geometries. 

The distance of a point from a line is the length of the perpendicular from the point to the line.