I don’t know what you mean by identical, but I’m not big on geometry, so I’ll explain my dumbed down misunderstanding of your statement.
If you mean each angle leaves its point of origin with the same degree relative to a line or axis drawn between them, then yeah, you have two congruent lines. But you can do this with any two angles leaving the same point of origin if you place a line directly between them.
Draw a line from the nut in the middle of the clock that bolts the hands down to the 12. There is your axis line. So each hand would have the same angle relative to the line. It would be a little more than a 45 degree angle for each hand, since the 3 and 9 would be a 90 degree angle while the 12 would be no angle.
No matter where you put the hands, you could draw a line directly between them, creating the same degree of each angle. That’s why I don’t understand what you mean when you say identical.
There is no such thing as an identical angle because all angles can be identical depending on the axis line between them.
Now if you insist that the 12 be the axis line, then putting the little hand on the 6 and the big hand on the 3, you would not have identical angles. The little hand would have a 180 degree angle while the big hand would have a 90 degree angle.
Really, mentioning that the angles are identical seems to be superfluous here.