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I have a staircase with trim above the skirtboard. At the top of the stairs the horizontal piece has been removed where the diagonal meets the horizontal and I am left with a currently unknown angle

What's the best way (without protractor, angle finder, etc) to figure out what angle I have to cut to meet the existing trim angle?

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All you have to do is measure from the top of your existing trim piece on the left straight across to the wall molding. Then measure from the bottom of the trim piece across to the same molding.
Take the piece of trim you want to piece in and first cut it at 90 degrees to fit against the molding on the right. Then mark off the correct measurements you took at the top and bottom of the trim starting from the end on the right.
Draw your diagonal and cut it. You can fill in any gaps with caulk or wood filler.

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So given you exclude the most obvious tools I would suggest cutting a card or cardboard template.

Even a cereal box can be suitable for this.

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