O sensei explains for ikkyo that keeping uke's am at a right angle is essential (page 41 of the English edition of Budo). But right with what? With uke's body? Yes, as one can see on O sensei's pictures, but that is only a basic level of reality.
Advancing to a more fundamental level of reality, the 90º angle is the angle of uke's arm with ikkyo's 23º entry angle.
One can't make a right angle with a bent side: if uke's arm makes a right angle, it is obviously because uke's arm is straight at the moment of the descent and lock (while it is not in the initial phase of the movement). Indeed, ikkyo only works if uke's arm is in extension, with a right angle when going down, it is only that way that the lever on his elbow is efficient.
The counterpart of that property is that maintaining uke's arm in a right angle extension is only possible if the entry is done in the 23º angle, the opening of the feet being in that case in the initial 60º direction.
Indeed if the rotation of the body is more important, the feet will open up more, until they reach the second position at 60º and that rotation of tori's body axis means uke's arm will be even more twisted and therefore can not be kept in textension. Further than the first 60º line means bending uke's arm, with the help of the characteristic nikkyo grab on the hand's back (Cf. O sensei's pictures below).
In that position where uke's arm is twisted, the necessary conditions for ikkyo are not gathered anymore for no lever can be applied on his elbow.
Because the arm is twisted, tori drive uke's elbow to the ground, the forearm being nearly vertical (a major difference with ikkyo), the consequence being that the wrist is locked above. That specific circumstance implies that nikyo's final immobilization is always done with that peculiar lock illustrated by the third picture below:
And that bio mechanical reason explains why the two locks are different. When uke's arm is bent in the descending phase, the moment for ikkyo is gone. That's when, and only when, nikyo's moment occur.
That descent of uke's arm by a wider rotation of tori's axis and the following immobilization are done in 60º wider angle than ikkyo omote's angle.
We can therefore put that fact on our figure:
And we can now add one more element to our figure:
Aikido is not a sport, it is a martial art which laws (takemusu) are in harmony with the laws of the universe. Studying them allows the practitioner to understand his place in the universe. Aikido was born in Iwama, O sensei achieved in that village the synthesis of tai jutsu, aiki ken and aiki jo.
The International Takemusu Aikido Federation (ITAF) brings to the practitioner the structure he needs in order to work as close as possible to the reality O sensei MU defined. The official national representations are the guarantee of a teaching faithful to the Founder's.
In modern Aikido, weapons are hardly taught, if taught at all. In O sensei's Aikido, on the contrary, aiki ken, aiki jo and tai jutsu are unified and form together a riai, a family of harmonious techniques stemming from one unique principle. Each techniques helps understand all the others.
Peace is a balance between a human being and the world around him. The true martial art's goal is not to become stronger than one's opponent but to find in that opponent a way to realize harmony. There is no enemy anymore as such, but an opportunity offered to reach unified ki.