"Question 6 In the following figure, if AC = BD, then prove that AB = CD.
Class 9 - Math - Introduction to Euclid's Geometry Page 86"
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Here, we use axiom 3 which states that equals are subtracted from equals then the remainders are also equal.
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Solution:
Given, AC = BD......(1)
From the figure,
AC = AB + BC
&
BD = BC + CD
On putting these values in eq 1,
⇒ AB +
BC = BC + CD
According to Euclid’s axiom, when equals are
subtracted from equals, remainders are also equal.
On Subtracting BC from both sides,
AB + BC – BC = BC + CD – BC
AB = CD [ by axiom 3 of Euclid]
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Hope this will help you....
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133
Euclid's Geometry.
_______________________
☛ Some of Euclid's definitions :
▪️A point is that which has no part.
▪️A line is Breadthless length.
▪️The ends of a line segment are points.
▪️A straight line extends indefinitely in both the direction.
▪️A surface is that which has length and breadth only.
▪️A plane surface is a surface which lies evenly with the straight lines on itself.
☛ Euclid's Axioms ;
▪️Things which are equal to the same thing are equal to one another.
▪️If equals are added to equals, the wholes are equal.
▪️If equals are subtracted from equals the, reminders are equal.
▪️ Things which coincide with one another are equal to one another.
▪️The whole is greater than the part
▪️Things which are double of the same thing are equal to one another.
▪️Things which are halves of the same thing are equal to one another.
☛ Euclid's 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 into 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.
_______________________
☛ Some of Euclid's definitions :
▪️A point is that which has no part.
▪️A line is Breadthless length.
▪️The ends of a line segment are points.
▪️A straight line extends indefinitely in both the direction.
▪️A surface is that which has length and breadth only.
▪️A plane surface is a surface which lies evenly with the straight lines on itself.
☛ Euclid's Axioms ;
▪️Things which are equal to the same thing are equal to one another.
▪️If equals are added to equals, the wholes are equal.
▪️If equals are subtracted from equals the, reminders are equal.
▪️ Things which coincide with one another are equal to one another.
▪️The whole is greater than the part
▪️Things which are double of the same thing are equal to one another.
▪️Things which are halves of the same thing are equal to one another.
☛ Euclid's 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 into 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.
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