Q. Give two equivalent versions of Euclid’s fifth postulates?
Solution: Two equivalent versions of Euclid’s fifth postulate are: (i) ‘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’. (ii) Two distinct intersecting lines cannot be parallel to the same line. Q. Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.
Solution. If a straight line l falls on two straight lines m and n such that the sum of the interior angles on one side of l is two right angles, then by Euclid’s fifth postulate the lines will not meet on this side of l. Next, we know that the sum of the interior angles on the other side of line l 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.
Q. If A, B and C are three points on a line, and B lies between A and C then prove that AB + BC = AC.
Ans: A, B and C are three points on a line Things which coincide with one another are equal to one another
AB + BC = AC. Q. Prove that an equilateral triangle can be constructed on any given line segment. Solution: Draw a line segment AB. Now draw two circles with centre A and B of radius AB. Then draw the line segments AC and BC to form Δ ABC Now, AB = AC, since they are the radii of the same circle (1) Similarly, AB = BC (Radii of the same circle) (2) From these two facts, and Euclid’s axiom that things which are equal to the same thing are equal to one another, you can conclude that AB = BC = AC. So, Δ ABC is an equilateral triangle
Q. Prove that two distinct lines cannot have more than one point in common.
Solution: Here we are given two lines l and m
If possible let the two lines intersect in two distinct points, say P and Q. So, you have two lines passing through two distinct points P and Q
But it is the axiom that only one line can pass through two distinct points. So, our supposition that two lines can pass through two distinct points is wrong. Hence, two distinct lines cannot have more than one point in common.
Q. write Some of Euclid’s axiom
Ans: Some of Euclid’s axioms were: (1) Things which are equal to the same thing 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 of the same things are equal to one another. (7) Things which are halves of the same things are equal to one another.
Q.A,B, C are three points on a line B lies between A and C , Prove that AB + BC = AC
Ans: AC coincides with AB + BC. Also, Euclid's Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AB + BC = AC