I’ve been tossing around terms like; erect a perpendicular, strike an arc; Geometry. Unlike what you may have thought in Middle School; Geometry is your friend.
The most frequently used method for erecting a perpendicular is by reference to a copy of a right angle  – Framing/Rafter Square, Speed Square, Combination Square, Try Square, Drywall Square, Miter Square. . .
However, there is another family of techniques – the  “compass and straightedge constructions.” that are extremely useful. Moreover, they often involve no measurement or calculation at all; and do not rely on whether or not your square is in fettle.
There’s a nice compilation at Math Open Reference with nifty animations and clear explanations.
These are the ones I used most often. Simplest first.
Bisect a line
Perpendicular to a line from a point
Erecting a perpendicular from a line at a stated point    
Drawing a Parallel line
The above is the one we usually used; but I like this one I just stumbled across better.

 

Quick and dirty method (4+ tangents – Euclid would not approve; but he might wink.)  

When striking an arc you can feel the scribing end move away from and back towards you. If you know the distance from the baseline to the parallel; set your compass to that distance and strike 4-5 arcs along the baseline – where you feel both the push and the pull. A straight line that kisses the extreme extents of all of those arcs (NOT THE POINTS WHERE THEY MAY, BY HAPPENSTANCE, INTERSECT) is parallel to the baseline at the stated distance.

Perpendicular gives you 90 degrees. You can also construct 60 and 30 degree angles

You can bisect an angle, very useful if you have to start from a given angle that isn’t readily measurable

You can add or subtract angles from one another to derive a whole assortment of useful angles.

How to find the center of a circle or an arc.

The perpendicular bisector of a chord lies along the radius of its arc. The point at which two or more of such radius lines (radii) intersect is the center of that arc.
The centering head on a Combination Square and the Stanley 46-101 Center Square both operate via the principle of the perpendicular bisector of a chord.

A line that connects the points at which the two limbs of a right angle (placed against a circle) intersect that circle, is a diameter of that circle. The point at which two diameters intersect one another is the center of that circle. It is a good practice to locate three or more, when doing this, to prevent errors. This method also works for circular arcs. However, you may not have enough arc to get a good read. If you are getting really funky results; you should suspect that you are actually in the presence of an elliptical arc. These are as different from one another as a shark is from a dolphin. They both swim in the ocean and eat fish; but that’s where the resemblance ends.

In a later post I will describe a solution for the reverse problem. How to draw an arc when you have no access to the center point.

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