Why do we need postulate 4 in Euclid's element (P14)?

[chemistry1]

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI14.html

Hi, I was reading proposition 14 of Euclid's elements and there is only one thing which I find weird : why do we need postulate 4 to conclude that " the sum of the angles CBA and ABE equals the sum of the angles CBA and ABD."

Why can't we just use common notion 1 ? It seems useless to me to use the postulate...

Thank you !

 

[AlephZero]

Euclid's definition of a right angle (definition 10) is a geometrical construction. You can't prove that right angles constructed at different positions are equal, so you need postulate 4 to say they are equal.

In modern notation, common notion 1 says "if x = a and y = a, then x = y". In the figure for proposition 14, if you call CBA+ABE x and CBA+ABD y, you don't have anything that corresponds to "a" in common notion 1.

Even if you constructed two more right angles somewhere in the figure and called then "a", you still need postulate 4 to say that x = a and y = a. But Euclid used postulate 4 directly to say that x = y.

 

[chemistry1]

Euclid's definition of a right angle (definition 10) is a geometrical construction. You can't prove that right angles constructed at different positions are equal, so you need postulate 4 to say they are equal.

In modern notation, common notion 1 says "if x = a and y = a, then x = y". In the figure for proposition 14, if you call CBA+ABE x and CBA+ABD y, you don't have anything that corresponds to "a" in common notion 1.

Even if you constructed two more right angles somewhere in the figure and called then "a", you still need postulate 4 to say that x = a and y = a. But Euclid used postulate 4 directly to say that x = y.

I don't think I understand your explanation... Didn't Euclid "prove" with proposition 13 that the sum of two angles were equal to two right angles ? And that we begin the demonstration assuming that the sum of the angle CBA and ABD were equal to two right angles ? What role does postulate 4 play here if I want to make things which equal the same thing equal one another ? What errors would I get if I just applied common notion 1 ?Thank you again for your help!