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I am installing some chair rail, and the points where it's a 90 degree connection (inside/outside) are no big deal... I cut 45s and then fit them together. But I have a couple of places where the angle is 130 degrees, so I have to cut at 65. For this stuff, I was using a hand saw and doing the cut, but I am wondering what the best way to handle these odd angles? Is there a trick or technique that could help me?

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The standard way to cut mitered corners is with a miter saw, either a hand saw with a miter box or a power miter saw.

I don't know how you're cutting your 45ºs currently. If you're using a hand saw and want good control, I would get a hand miter box with an adjustable angle, like this one on amazon. Usually they have stops at common angles like 45º and 22.5º, but you can set it to any angle (I don't have any experience with this model though): enter image description here

If you want something quick and dirty I would just use a carpentry protractor / angle finder to draw a line and then freehand it:

enter image description here

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