In the animation above, the green and purple quadrilaterals go through three stages, which I call "square", "parallelogram", and "rectangle". I hope this isn't confusing; after all, every square is a rectangle, and every rectangle is a parallelogram. When I say "rectangle" in these questions, I mean the non-square ones; and when I say "parallelogram", I mean the non-rectangular ones that have an edge along the line drawn through C.
At three times in the animation, Triangle ABC is rotated by some amount. What are the measures of the three angles of rotation?
All three rotations are by 90 degree angles.[br][br]The first rotation sends AC to another side of the green square; the second rotation sends BC to another side of the purple square; the third rotation sends AB to another side of the orange square.[br][br]Alternate answer: if you want a sign to indicate direction, then I would say the first and third rotations are by positive 90 degrees (counter-clockwise) while the second rotation is by negative 90 degrees (clockwise). Reasonable people can disagree about (1) which direction should be considered positive, and (2) how angles should be measured (e.g. degrees or radians), so answers may vary.
The green square is transformed into a parallelogram. What reason is there to think that this parallelogram has the same area as the original green square?
If C is closer to B than it is to A, then one answer is that the square and the parallelogram overlap on a dark green trapezoid, and each of them covers a triangle that the other doesn't. The square has the same total area as the trapezoid and one triangle together. The parallelogram has the same total area as the trapezoid and the other triangle together. Since the two triangles have the same area, so do the square and trapezoid. How do we know that the two triangles have the same area? That's sub-question 2a below.[br][br]A second answer is that the way the square is being transformed is called a shear transformation, and this kind of transformation always preserves area.
Make sure C is closer to B than to A. Pause the animation while the purple square is being transformed into the purple parallelogram. At this moment, how can we know that the two green triangles have the same area?
One way: opposite sides of a parallelogram have the same length. The square and the parallelogram--both of which are parallelograms--share side AC, so their sides opposite AC are congruent. Subtracting off their overlap, the two remaining ends become congruent bases for the two triangles.[br][br]Meanwhile, the heights of both triangles are sides of the same green square, so they must be congruent.[br][br]We can apply the triangle area formula now, or point out some right triangles. By SAS, the triangles are congruent. There are other ways to make this case.
The green and purple parallelograms are transformed together into a pair of adjacent rectangles. How do we know that these rectangles have the same areas as the matching parallelograms they came from?
One reason is that both parallelograms are being acted upon by (separate) shear transformations, which preserve area.[br][br]You may be able to find another explanation.
How do we know the green and purple rectangles combine to make a square, and that this square has the same area as the orange square?
Their combined width is side AB, which is the hypotenuse of triangle ABC. Meanwhile, each rectangle's longer side is the hypotenuse of a triangle congruent to ABC, so we have three sides congruent.[br][br]Each square was only ever acted upon by shear transformations, and these always send parallel lines to parallel lines. Therefore, the shapes I've been calling rectangles really are rectangles: they are parallelograms with a right angle. Rectangles with a common side form a third rectangle, and this one (as mentioned in the previous paragraph) has congruent adjacent sides. It is therefore a rhombus; but a rectangle that is also a rhombus is a square.[br][br]Finally, the green-and-purple square shares side AB with the orange square, so these two large squares have equal areas.