Therefore, provided that you are willing to at least entertain for the moment that it's the SUN shining on the MOON and us looking at the MOON from EARTH that's the reason that we see a quarter moon -- that is, the moon as a globe, or spherical object, with light shining on its right side, while being almost entirely dark on its left side (in the northern hemisphere, obviously, USA or Canada), with the line separating the light side and dark side going somewhat vertically from the top of the moon to the bottom -- then we can move on to the next aspect of this observation, which is: measurement of the angle between two different lines -- our line of sight toward the sun and our line of sight toward the moon.
The link I provided above gives the time of day from the Los Angeles area as 11:40 am, when the first quarter moon will occur. That is the preferred time to make the observation I will describe below. It is of no great consequence if the observation is made in my area later in the day, for example one or two o'clock in the afternoon, or even until sunset, which occurs here at 7:16 pm. The time of the observation should be noted, however, so it can be used in the analysis that follows.
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The first quarter on April 3rd was at 11:40 am. That put the moon below the horizon where we can't see it yet, but it was moving upwards. Maybe that's where you got the idea of "moonrise", mw.
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The next opportunity on April 19th will be at 3:00 am and it will be the last quarter, which means the moon will lead the sun. The moon will already be high in the sky (can you predict how high??) when the sun rises in the eastern horizon. People in the eastern USA will be able to see the moon at that time because the last quarter moon optimum time will occur at 6:00 am there (which is objectively the same time as 3:00 am PDT), with sunrise at 16 minutes later, at 6:16 am. So they'll be able to measure the sun-moon angle less than a half-hour after the optimum viewing time. Even though it is a few minutes late when they'll have the ability to measure the sun-moon angle, it will be a better time to get this angle, than our time will be on the west coast, when the sun rises 3 hours later. For us, we'll be over 3 hours late, but in the Carolinas, they'll only be 16 minutes late.