Chapter-3: Introduction to Euclid’s Geometry

Exercise – 3A

Q1: What is the difference between a theorem and an axiom.

Sol: A theorem is a statement that requires a proof. Whereas, a basic fact which is taken for granted, without proof, is called an axiom.

Example of Theorem: Pythagoras Theorem
Example of the axiom: A unique line can be drawn through any two points.  

Q2: Define the following terms

(i) Line segment

(ii) Ray

(iii) Intersecting Lines

(iv) Parallel Lines

(v) Half-line

(vi) Concurrent lines

(vii) Plane

Sol:

(i) Line segment: The straight path between two points is called a line segment.

(ii) Ray: A line segment when extended indefinitely in one direction is called a ray.

(iii) Intersecting Lines: Two lines meeting at a common point are called intersecting lines, i.e., they have a common point.

(iv) Parallel Lines: Two lines in a plane are said to be parallel, if they have no common point, i.e., they do not meet at all.

(v) Half-line: A ray without its initial point is called a half-line.

(vi) Concurrent lines: Three or more lines are said to be concurrent if they intersect at the same point. (vii) Collinear points: Three or more than three points are said to be collinear if they lie on the same line.

(vii) Plane: A plane is a surface such that every point of the line joining any two points on it, lies on it.

 

Question 3: In the adjoining figure name

(i) Six points:

(ii) Five line segments:

(iii) Four rays:

(iv) Four lines:

(v) Four collinear points:

Sol:

(i) Six points: A,B,C,D,E,F

(ii) Five line segments: EG, FH, EF, GH, MN

(iii) Four rays: EP, GR, GB, HD

(iv) Four lines: AB, CD, PQ, RS

(v) Four collinear points: M,E,G,B

 

Q4: In the adjoining figure name:

(i) Two pairs of intersecting lines and their corresponding points of intersection.

(ii) Three concurrent lines and their points of intersection.

(iii) Three rays.

(iv) Two line segments.

Sol:

(i) (EF GH) and their corresponding point of intersection is R.

(AB CD) and their corresponding point of intersection is P.

(ii) (AB EF GH) and their point of intersection is R.

(iii) Three rays are: RB, RH, RG

(iv) Two line segments are: RQ, RP

 

Q5:

(i) How many lines can be drawn to pass through a given points?

(ii) How many lines can be drawn to pass through two given points?

(ii) In how many points can the two lines at the most intersect?

(iv)  If A,B,C are three collinear points, name all the line segments determined by them.

Sol:

(i) An infinite number of lines can be drawn to pass through a given point.

(ii) One and only one line can pass through two given points.

(iii) Two given lines can at the most intersect at one and only one point.

(iv) AB, BC, AC

 

Q6: Which of the following statements are true

(i) A line segment has no definite length.

(ii) A Ray has no end point.

(iii)  A line has a definite length.

(iv) A line AB is the same as line BA.

(v) A ray AB the same as ray BA.

(vi) Two distinct points always determine a unique line.

(vii) Three lines are concurrent if they have a common point.

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

(ix) Two intersecting lines cannot be both parallel to the same line.

(x) Two lines may intersect two points.

(xii) Two lines l and m are parallel only when they have no point in common.

Sol:

(i) False

(ii) False

(iii) False

(iv) True

(v) False

(vi) True

(vii) True

(viii) True

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