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The sun traverses the sky faster than the moon does. So every hour or two, the moon moves some number of degrees further away from the sun after the new moon and before the full moon.

After the full moon, the sun starts to catch up to the moon and its approach makes the angle between their respective lines of sight from earth to get smaller, until at the moment of new moon, they are both in the same location in the sky, from the reference point of earth with the north star behind the viewer.

Now that I have done this once, I see that it's a good idea to check the angle between the sun and moon the day BEFORE the quarter moon, noting the clock time when checking the angle. Then if you can again check the angle the following day at the same number of minutes after the Almanac shows the time of the quarter moon as your previous day's measurement time was BEFORE the same Almanac time, then you can estimate what the angle would have been if you had checked it at precisely the Almanac time.

There will be times when measuring this angle at the moment of the Almanac time will not be possible, such as it was today. If you had to wait until two or three hours after moonrise, the sun was then in the afternoon sky, since it was about 3 or 4 o'clock P.M.

The last quarter of April will be in 2 weeks, which will occur at 3:00 am on April 19th:

Old Farmer's Almanac moon phases for month of April from Santa Monica

Well, obviously, you won't be able to see the moon at 3 o'clock in the morning. Nor will you be able to see the sun, since it will be night time.

The first quarter moon follows the sun, and the second quarter moon leads the sun.

The best plan is to look for a time when the moon and sun will both be visible in the sky at the moment of the quarter moon according to the Almanac. For the first quarter, look for a time from 1 pm to 7:30 pm (daylight savings time), a span of 6 or 7 hours. For the last quarter, look for an Almanac time from 6 am to 4 pm. Therefore, this measurement is more easily made during the last quarter, when the moon leads the sun across the sky, because you'll be able to see both of them from sunrise until the moon sets 10 hours later. But you'll have to do it in the first hours of daylight until mid-afternoon. If you can better find the time in the early evening, you should shoot for a first quarter moon viewing.

The last quarter moon of this past January was at 2:14 pm, which would have been a good time, since the moon would be close to the western horizon but still visible.

You should have posted this before yesterday, if you wanted me to look at it and measure the angle between the moon and sun.
I can do it today because the moonrise time is only a few moments later.
But, you still have not explained what is the point of this experiment? Just get to the point. What is it you think can be shown by measuring the angle between moon's position at rise and where the sun is in relation to it at that moment?

You should have posted this before yesterday, if you wanted me to look at it and measure the angle between the moon and sun.
I can do it today because the moonrise time is only a few moments later.
But, you still have not explained what is the point of this experiment? Just get to the point. What is it you think can be shown by measuring the angle between moon's position at rise and where the sun is in relation to it at that moment?

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It's not moon's position at rise. You keep saying that and I have no idea where you got it. 
We're not concerned with moonrise, UNLESS that is the moment when the first or last quarter moon occurs that day. I have given you links to the Almanac website that displays the times for each quarter moon but maybe you think that's part of the great conspiracy or whatever. You're going to have to get OVER that prejudice if you want to observe the facts without prejudice.
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I know, I should have posted it before yesterday. But what is, what is. Now we'll have to wait for the next opportunity. A measurement can be taken on May 1st in the evening, and two more on May 2nd in the morning and in the evening. The first quarter happens at 7:48 pm which is 9 minutes after sunset (7:39 pm). So the sun will be very low in the sky, and the moon will be somewhere overhead. --- Can you predict about where in the sky the moon will be on May 2nd when the sun is setting? 
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The sun, moon and planets make regular motions in the sky, which allow us to predict where they will be, even many years in advance. Can you explain how that would be possible with a "flat" earth model? Are there any astronomical charts and tables written by flat-earthers? For example, if a viewer in California measures the angle between the moon and sun as 93 degrees like I did, and another viewer in Hawaii sees the same 93 degrees and another viewer in Quito, Equador and another in Calgary, Alberta, Canada sees 93 degrees and another in Houston TX also sees 93 degrees, all of them at the same hour (5:00 PDT, or UTC -7), how would you explain that set of data using a flat-earth model? But in any case, there is a lot more to this process, but it would only bog down the topic at this point. Go watch the video if you want the whole process.
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Measurements taken today are not of no use, but they are unable to show the key picture that we need for the next phase. We need to see what the angle is between the moon and the sun as viewed from earth at the moment of the quarter moon. And that moment has passed for this phase. The next quarter will be the LAST quarter moon, on April 19th. Remember, during the last quarter, the moon leads the sun, so the moon will be about 6 hours ahead of the sun. The last quarter occurs at 3:00 am, therefore the moon will be around the 9 o'clock position of a 24-hour clock, which is 10:30 on a 12-hour clock. Since the sun moves faster than the moon, the sun will be catching up to the moon that morning on Wednesday the 19th of April. If we get a chance to measure the angle between them as the sun rises, it is going to be a smaller angle than it was at 3:00 am, but how much smaller is another question.
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Today the area of the illuminated moon has increased 11% from yesterday, from 50% to 61%. So we can see that the angle of the sun to the moon is significantly changing. This has many implications.
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The point of the experiment is to complete the picture of the relative positions of the earth, moon and sun at the moment of the first or last quarter moon each month. This position occurs 24 times each year, so the data is very available for those who want to know what reality has to show us. I already posted a video that goes through the whole process if you want to watch that again -- or, perhaps for the first time, if you have not yet watched the videos I've posted.  
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I make it a point to watch the F-E-er's videos all right, but they don't bother to watch mine, apparently. Because if they had watched them they wouldn't be asking these questions over and over again. Maybe it's too much work to spend 20 minutes every few days or whatever. 
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