Editing The Road to Reality Study Notes (section)

===2.3 Similar-areas proof of the Pythagorean theorem===
Penrose revisits the Pythagorean theorem by outlining another proof. Starting with a right triangle, subdivide the shape into two smaller triangles by drawing a line perpendicular to the hypotenuse through the right angle. The two smaller triangles are said to be ''similar'' to one another meaning they have the same shape but are different sizes. This is true because each of the smaller triangles has a right angle and shares an angle with the larger triangle. The third angle known because the sum of the angles in any triangle is always the same. Knowing that the sum of the area of the two small triangles equals the area of the big triangle (by construction), we can square the sides and show that the pythagorean theorem holds. 

Again Penrose asks us to revisit our assumptions and examine which of Euclid's postulates were needed. Particularly our claim that the sum of the angles in a triangle add up to the same value of 180° (or <math>\pi</math> [https://en.wikipedia.org/wiki/Radian radians]). One must use the parallel postulate to show that this is true. Penrose asks us to consider what would it mean for the parallel postulate to be false? What would that imply? Would that make any sense? With these questions in mind we begin to explore a different kind of geometry.