Many of the theorems Euclid proved were known long before he wrote Elements. For example, Thales knew that the angles in a triangle summed to a straight angle (180 degrees) and that the base angles of an isosceles triangle were congruent. He also knew that an angle that subtends the diameter of a circle must be right, as shown below.
In the figure shown above, angle C subtends diameter AB. Therefore, angle ACB must be right.
Euclid proved the theorem formally by drawing the radius from O to C, and noting that the segments OA, OB, and OC are all congruent, as radii of the same circle. Thus, the large triangle is broken up into two isosceles triangles--one with a pair of base angles that measure X, and the other with a pair of base angles that measure Y. Because 2x+2y form the interior angles of the original triangle, they must sum to π. Thus x+y=π/2.
https://mathcs.clarku.edu/~djoyce/elements/bookIII/propIII31.html
However, there is a more intuitive and essential argument to be made, if we focus on the mental actions that define geometric figures (cf., Felix Klein's Erlangen Program).
Note that if we extend a segment from C, through circle center O, to point D, we have a second diameter of the circle. The two diameters bisect each other at O and form the diagonals of a quadrilateral. We know that only rectangles have this property, and that all of its angles are right. However, we can make a more dynamic argument.
The circle is the geometric figure with perfect symmetry: it remains invariant under transformation by any rotation through its center or by any reflection about a line through its center. These form a group of transformations. The two diagonals define a quadrilateral with a subgroup of thee transformations/symmetries: a 180 degree rotation and two lines of reflection. These are the symmetries that define a rectangle, and the same symmetries that guarantee it has four right angles.
Euclid proved the theorem formally by drawing the radius from O to C, and noting that the segments OA, OB, and OC are all congruent, as radii of the same circle. Thus, the large triangle is broken up into two isosceles triangles--one with a pair of base angles that measure X, and the other with a pair of base angles that measure Y. Because 2x+2y form the interior angles of the original triangle, they must sum to π. Thus x+y=π/2.
https://mathcs.clarku.edu/~djoyce/elements/bookIII/propIII31.html
However, there is a more intuitive and essential argument to be made, if we focus on the mental actions that define geometric figures (cf., Felix Klein's Erlangen Program).
Note that if we extend a segment from C, through circle center O, to point D, we have a second diameter of the circle. The two diameters bisect each other at O and form the diagonals of a quadrilateral. We know that only rectangles have this property, and that all of its angles are right. However, we can make a more dynamic argument.
The circle is the geometric figure with perfect symmetry: it remains invariant under transformation by any rotation through its center or by any reflection about a line through its center. These form a group of transformations. The two diagonals define a quadrilateral with a subgroup of thee transformations/symmetries: a 180 degree rotation and two lines of reflection. These are the symmetries that define a rectangle, and the same symmetries that guarantee it has four right angles.