Complete script

So, yeah, I will just give you the script I got so far.

1.1.2 Spoken text by miss jane
Alright then—so, what exactly is the Pythagorean Theorem?
The Pythagorean Theorem is a mathematical rule that connects the side lengths of a right-angled triangle.
To make things easier, let’s give names to the sides of the triangle.
We’ll call this side: leg a.
This side: leg b.
And this one: the hypotenuse, which we’ll call c.
Now, the Pythagorean Theorem tells us mathematically how these three sidelengths are connected in every right-angled triangle.
Meaning, if we know two of the side lengths—let’s say we know the length of leg a and the length of leg b—we can use the Pythagorean Theorem to calculate the length of the hypotenuse c, without actually having to measure it.
It's a rule that uses the fact that the third side length is already determined as soon as you chose two.

1.1.3 Spoken text by Ms. Jane
In other words if 2 sidelengths of a rectangular triangle are chosen, the third one is no longer variable, its no longer free, it is automatically determined by the first two.
now one might ask, why should the third sidelength of a rectangular triangle be fixed as soon as 2 are given and to answer that legitimate question, i prepared a visual that should make that clear.

1
voice: so, in general given two sidelengths, lets say this is leg a and this is leg b for a triangle, there are infinite possibilities for the third sidelength of c.
visual: first draw a certain length of a horizonzal line in green and add the label leg a below the line, also in green, then do the same thing for another vertical line in blue, also with label, but this label is left to the line, then joint the left end of leg a, with the lower end of leg b, then draw a line from the right ending point of leg a, to the upper ending point of leg b, this line should be in orange and outside so above and at the right of this line, you add the label c, also in orange.
2
voice: to illustrate that given the length of leg a and leg b, length of c is variable we can rotate our legs away from each other, this way length of leg a and b remains the same whilst length of c grows, or we can rotate our legs towards each other, to make c smaller.
visual: wiggle first leg a and then leg b, then change the length of c, whilst maintaining all joints between the 3 lines, and maintain the length of leg a and leg b, but adjust its angle so give up the rectangularity between leg a and leg b, make c first longer and then shorter and then go back to the initial rectangular configuration
3
voice: however i told you that the pythagorean theorem is a rule that applies for a rectangular triangle and as you can see if we constrain the angle between these sides to be rectangular, there is only one possibility for the lenght of c. If we made c longer, we either would keep leg a and leg b and thereby no longer having a triangle or we would keep the joints and thereby breaking the rectangularity condition. So, if we keep legs a and b fixed, and we don’t break the triangle or the right-angle constraint, then there’s only one possible length for the third side to complete the right triangle.
visual: wiggle leg and leg b simultaneously, then add a rectangle symbol between leg a and leg b, then leave all angles equal but make c longer in both directions, such that the joint constraint between right end of leg a and lower end of c is broken and the same for the joint between upper point of leg b and upper point of c, so leg a and leg b should be unaffected but make c first longer and then bring it back to its original length with the joints with leg a and leg b, then make c again longer, but this time you make c longer, the way you did in 2, using the rotation of leg a and b, away from each other, then bring leg a and b back into rectangular original configuration.

Aight, uh… this might sound dumb but—okay—so last summer I was in Rome, with my mom and We did this tour, and the guy kept talking about these arch technology used to build bridges and viaducts. Like old stone ones.
And he said—this blew my mind—in the roman empire they built those arches without glue. Like no cement, no mortar, nothing. Just… rocks piled on top of each other with the right shape and angles and they had a bridge, that could carry tons.
Visual: show an image of a roman bridge
Now, imagine you are a roman empire architect and you have to build a bridge over a river at a given location, using 3 stones and that technique the romans used.
then the same logic applies as you explained:
keystone arch
pythagorean theorem
bridge
triangle
riverbeds shape
rectangular
shape of sidestoneleft
length of leg a
shape of sidestoneright
length of leg b
keystones shape
hypothenuses length
the task is to build a bridge, whilst in the pythagorean therem, the task is to build a triangle.
so for the pythagorean theorem you said that given a rectangular triangle and 2 known sidelengths of it, the third sidelength is already determined. here i would say the same principle applies: given a fixed riverbed shape and two of the 3 stones you use to build the bridge, there is only one possible shape for the third stone to complete the assembly into a bridge.
The third stone’s shape isn’t a choice. It’s a consequence.
you know, add some stones and bamm, roman bridge done!

1.1.10

ayla(sharp): (Maybe switch to manim) The rule that gives us the correlation of the triangle angles. It sais that the three angles of any triangle always add up to 180 degree.
so lets draw a trinagle, this rule applies to all triangles, not only rectangular ones, so any triangle, lets call this angle alpha, this beta and this gama, now the rule tells us that alpha + beta + gamma = 180°, this means if we know alpha and beta, we can use this rule to determine gama, without actually measuring gamma, same concept isnt it, given a triangle of which we know two angles we can determine the third one, using the rule, or as rio said it, The third angle isn’t a choice. It’s a consequence of the first two.