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Q1. Give the equivalent version of Euclid's fifth postulate in terms of parallel lines.

Solution

'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'.

Q2. Define : (A) a square (B) Perpendicular lines

Solution

Square - A square is a rectangle with a pair of consecutive sides equal. Perpendicular lines - Two lines are perpendicular, if the angle between them is 90^o.

Q3. Write any three Euclid's postulates.

Solution

Any three of these five postulates Postulate 1: A straight line may be drawn from any one point to any other point. Postulate 2: A terminated line can be produced indefinitely. Postulate 3: A circle can be drawn with any centre and any radius. Postulate 4: All right angles are equal to one another. Postulate 5: 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.

Q4. Explain how a Theorem is different from an Axiom.

Solution

Axioms are assumptions, which are taken for granted used throughout mathematics. Whereas, Theorems are statements that require proof.The process of establishing the truth of a statement is its proof.

Q5. Which of the Euclid's postulates implies the existence of parallel lines? Also, state the postulate.

Solution

Euclid's Fifth postulate implies the existence of parallel lines It states that:- 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.

Q6. Define: (A) line segment (B) radius of a circle

Solution

(A) A line segment is a part of line between two points. (B) Radius of a circle is defined as the distance between center and a point on its circumference.

Q7. Write any five Euclid's axioms.

Solution

Any five of these seven Euclid's axioms (a) Things which are equal to the same thing are equal to one another. (b) If equals are added to equals, the wholes are equal. (c) If equals are subtracted from equals, the remainders are equal. (d) Things which coincide with one another are equal to one another. (e) The whole is greater than the part. (f) Things which are double of the same things are equal to one another. (g) Things which are halves of the same things are equal to one another.

Q8. What are axioms and postulates ?

Solution

Axioms and postulates are the assumptions which are obvious universal truths. They are not proved.

Q9. Show that if two circles are equal then their radii are equal.

Solution

One of Euclid's axioms states that: Things which coincide with one another are equal to one another. If two circles are equal then the circles coincide i.e their centres coincide and their boundaries coincide. Therefore, their radii are equal

Q10. Euclid's axiom 4 is: Things which coincide with one another are equal to one another. What is the implication of two (i) lines (ii) circles (iii) triangles coinciding ?

Solution

(i) The implication of two lines coinciding is that the lengths of the two segments are equal.(ii)The implication of two circles coinciding is that their radii are equal.(iii) The implication of two triangles coinciding is that they are equal in all respects.