Introduction To Euclid's Geometry

(i) Given two points P and Q on a line the segment of line with end points P and Q, is called the line segment PQ. The line segment PQ has fixed length. Line segment PQ is denoted by PQ.

(ii) Three or more points P, Q and R are said to be collinear if they lie on a single straight line .

(iii) Parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel lines. Parrallel lines are denoted by

(iii) Two lines are intersecting if they have a common point. The common point is called the point of intersection.

(iv) When three or more lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point.

(v) A ray is a line with a single endpoint (or point of origin) that extends infinitely in one direction.

(vi) If P, Q and R be the points on a line , such that Q lies between P and R, and if we remove the poiny Q from the line the two parts of the line that remains are each called a half line.

Infinitely Many

One point only

One

Line segments are:

(i) True

(ii) False

Two lines intersect at one point only.

(iii) False

A line segment has fixed length.

(iv) True

(v) False

(vi) True

(vii) False

A segment has two end points

(viii) False

(ix) False

Infinite number of lines can pass through a point

(x) False

Two coincidenct lines have infinite number of points in common

(i) Five line segments are:

(ii) Five rays are: ray QC, ray PM, ray RB, ray DF, ray LH

(iii) Four Collinear points are: PRQ, PTN, QLD, NSD

(iv) Two pairs of non-intersecting line segments are: PN, RS and PQ, TL

(i) Unique

(ii) Lines

(iii) Perpendicular; Perpendicular

(iv) Three, two half planes, line

Two

At least two numbers of distinct points determine a unique line.

One

As shown above only one line can be drawn through both of the given points.

Infinitely many

As shown above from a given point infinite lines can be drawn.

One

As shown in the fig only at one point two distinct lines can intersect.

One

If a line intersects a plane that does not contain it, then it intersects the plane in exactly one point.

Infinite

Two planes always intersect in a line if they are not parallel. Therefore there will be infinite points of intersection for two planes.

One

It is this property which makes the plane "flat." Two distinct lines intersect in at most one point whereas two distinct planes intersect in at most one line. If two coplanar lines do not intersect then they are parallel.

Three non-collinear points

A set of three non-collinear points determine a unique plane.

One if they are collinear and three if they are non-collinear

For Collinear Points:

For three non-collinear points:

As shown in the above figures only one line can be drawn if three points are collinear and three lines can be drawn if three points are non-collinear.

One

Only one plane can be made to pass through a line and a point not on the line.

Infinite

Any two distinct points in three dimensional spaces determine a unique line in three dimensional space. This line has infinitely many planes that contain it. And also the two given points.

Infinite if they are collinear and only one if they are non-collinear

Infinite planes can pass through three collinear points and only one plane can pass through three non-collinear points.