Your rising sign is the sign that was climbing over the eastern horizon at the minute you were born. People who study charts call it the ascendant. People who live in a chart just call it the door. It is the way you walk into a room and the impression you leave when you walk back out, and it is often nothing like your sun.
Why the rising can feel more true than the sun
Because it is the first thing other people see, and after a while you start to believe it. A Scorpio sun with a Sagittarius rising will be read as bright, traveled, easygoing for years, and slowly become a little more of that on purpose. The rising is the costume that becomes a habit.
Why the exact birth time is critical
The rising changes every two hours. Born at 5:55 vs 6:05, you can have completely different rising signs. If you do not know your birth time, you can still get a useful chart, you just hold the rising more loosely. Asking your mother, or pulling the birth certificate, is one of the most worthwhile hours you can spend on your own chart.
What the rising does in a relationship
It is what your partner first fell for and what acquaintances think they know about you. When the relationship gets older, the sun and moon start running the show, and the partner sometimes feels confused. They were sure they were dating one person. They are now dating someone else, on purpose. That is the rising stepping back and the rest of you stepping forward.