1. If a point C lies between two points A and B such that AC = BC, then prove AC = ½ AB. Explain by drawing the figure :

AC = BC
AC + AC = BC + AC [ Equals are added to equally.
2AC = AB [ BC + AC coincides with AB ]
AC = ½ AB.

2. If a point C lies between two points A and B such that AC = BC, then prove that every line segment has one and only one mid – point.
Let a line AB have two mid – points say C and D then
AC = ½ AB ............ (1 )
AD = ½ AB ............ (2 )
From (1) and (2)
AC = AD [Things which are equal to the same thing are equal to one another.]

3. In figure 1 if AC = BD, then prove that AB = CD

AC = BD            ........... (1) [given
AC = AB + BC   ........... ( 2 ) [ point B lies between A and C ]
BE = BC + CD   ........... (3 ) [ point C lies between B and D ]
Substituting (2 ) and (3) in (1 ), we get AB + BC = BC + CD
AB = CD [ Subtracting equals from equals.]

4. Does Euclid’s fifth postulate imply the existence of parallel lines ? Explain.
      If a straight line l falls on two straight lines m and n such that sum of the interior angles on one side of R is two right angles, then by Euclid’s fifth postulate the line will not meet on this side of next, we know that the sum of the interior angles on the other side of line will also be two right angles Therefore, they will not meet on the other side also. So, the lines m and n never meet and are, therefore parallel.

5. State Euclid’s 5 postulates :
1. A straight line may be drawn form any one point to any other point.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any centre and any radius.
   All right angles are equal to one another.
5. If a straight line falling on two straight lies makes the interior angles on the same side of it taken together less than two right angles, then the straight lines, it produced indefinitely meet on that side on which the angles are less than two right angles.

6. Which of the following statements are true and which are false ? Give reasons.
A. Only one line can pass through a single point
B. There are an infinite number of lines which pass through two distinct points
C. If two circles are equal, then their radii are equal

A. False, this can be seen usually.
B. False, This contradicts Axiom : Given two distinct points, there is a unique line that passes through them.
C. True, If we superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide therefore, their radii will coincide.

7. Consider two ‘postulates’ given below :
Do these postulates contain any undefined terms ? Are these postulates consistent ? Explain.
( i ) Give any two distinct points A and B, there exists a third point C which is in between A and B.
(ii ) There exist at least three points that are not on the same line.

Yes, These postulates contain two undefined terms : point and line.
Yes , These postulates are consistent because they deal with two different situations.
( i ) Say that given two points A and B, there is a point C lying on the line in between them.
( ii ) Say that given A and B, we can take C not lying the line through A and B. These postulates do not follow from Euclid’s postulate however they follow from Axiom 5 ie) “ Given two distinct points, there is a unique line that passes through them”.

8. Give two equivalent versions of the Euclid’s fifth postulate.
Two equivalent versions of the Euclid’s fifth postulate are :
( i ) for every line ∫ and for every point P not lying on  ∫ , there exists a unique line m passing through P and parallel to  ∫ 
( ii ) Two distinct intersecting lines cannot be parallel to the same line.

9. Explain Euclid’s Axioms.
1. Things which are equal to the same things are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal
4. Things which coincide with one another are equal to one another.
5. The Whole is greater than the part.
6. Things which are double than the part
7. Things which are halves of the same thins are equal to one another.

10. Define the following terms.
     (i) Parallel lines         (ii) Perpendicular lines

       (i) Parallel lines :- Two lines ,in a plane ,are said to be parallel,If they have no point in common .The distance between two parallel lines is always the same .
      (ii) Perpendicular lines :- Two lines, in a plane, are said to be perpendicular, if the angle between them is 90⁰.

11. Distinguish between a theorem and an axiom.

Axioms / Postulates : These are the assumptions which are obvious universal truth  and so they need not be proved .Axioms and postulates are the terms that  are used inter   changeably and in the same sense.
Theorems: These are statements which are proved,using definitions, axioms / postulates, previously proved results and deductive reasoning.

12. If AB is a line and P is a fixed point outside AB,how many lines can be drawn through P
      Which are (i) Parallel to AB              (ii) not parallel to AB ?

      
       (i) We can draw infinite number of lines parallel to AB. But only one Parallel line to AB.
       (ii) We can draw infinite number of lines through P which are not parallel to AB.

16. Define Collinear points.

        Three or more points are said to be collinear if there is a line which contains all of them.

17. Define concurrent lines.

        Three or more lines are said to be concurrent if there is a point which lies on all of them .

18. Complete the following statements with an appropriate word/term to be filled in the blank space(s).

1.______are the assumptions which are obvious universal truths.

2. Things which are equal to the same thing are _______ to one another.

3. If equals are subtracted from equals, the remainder are _____

4. Things which are double of the same things are ______ one another.

5. Two distinct intersecting lines cannot be _____ to the same line.

1. Axioms

2. equal

3. equal

4. equal

5. parallel

19. Read the following statements and write your answer as true or false.

1. In Fig., if AB = PQ = XY, then AB = XY

2. Only one line can pass through a single point.

3. If two circles are equal, then their radii are equal.

4. A terminated line can be produced indefinitely on both the sides.

5. There are an infinite number of lines which pass through two distinct points.

1. True

2. False

3. True

4. True

5. False