Copy lengths and angles with a proof

Try

A technician needs to transfer the length of a diagonal brace onto a new ray, then reproduce its joining angle. A ruler marked in centimeters can make an approximate measured drawing. An ideal compass-and-unmarked-straightedge construction instead gives a finite sequence of circles, lines and selected intersections whose correctness follows from geometry. Neither method becomes exact merely because a computer displays many decimal places.

We permit a compass that transfers a chosen distance and an unmarked straightedge that draws through two distinct known points. Paper folding will supply a second tool method. All folds are ideal reflections; real paper thickness, compass slip and line width create measurement error. State those distinctions when reporting a physical construction.

Predict

Suppose an angle is represented by two rays from A. Mark B,C on them at the same distance r from A. Predict whether transferring chord BC to another circle of a different radius must preserve the angle. Then predict how many candidate second rays can be obtained when the correct radius is used. Decide which extra instruction chooses the intended side.

Experiment

Carry out these instructions on paper. First copy a nonzero segment UV onto a given ray from P: set the compass to UV and draw its circle about P. Its unique intersection Q with the ray gives PQ=UV. Do not use the intersection on the opposite half-line.

For angle copying, take a source angle strictly between 0 and 180 degrees. Draw a circle about A meeting the two rays at B,C. At the target vertex P, draw a circle with the same radius and mark Q on the given starting ray. Draw a circle centered at Q with radius BC. Choose its intersection R with the P-circle on the requested side of PQ, then draw ray PR.

Object transferred	Construction operation	Equality obtained
Source radius AB	Circle about P	PQ=AB
Same radius AC	Same P-circle	PR=AC
Chord BC	Circle about Q	QR=BC

Trace the source and target triangles with matching labels. SSS proves angle QPR equals BAC. For a concrete source with r=5 and a 60-degree angle, BC=5 as well: both center triangles are equilateral. Changing the target radius to 8 while retaining chord 5 would produce a different triangle and a different central angle.

Understand

Why do the intended intersections exist? A nonstraight, nonzero source angle gives a chord satisfying 0<BC<2r. Two circles with radii r and BC whose centers are distance r apart therefore meet in two distinct points, symmetric about PQ. Selecting the side removes that ambiguity. At a zero angle the chord collapses; at a straight angle it becomes a diameter, so the generic two-intersection instruction must be replaced by the appropriate coincident or opposite ray.

To bisect an angle, mark B,C at equal positive distances on its rays. Construct the perpendicular bisector of BC using equal circles about B,C with radius greater than BC/2; join their two intersections. Since AB=AC, this bisector line contains A. Choose its ray inside the angle. For any other point X on that ray, XB=XC, while AB=AC and AX is common. SSS makes triangles ABX and ACX congruent, proving the two subangles equal. The selected ray, not the opposite extension, is the internal bisector.

Paper folding gives an alternative: fold one boundary ray onto the other with their common vertex fixed. Reflection in the crease exchanges the rays and preserves angles, so its internal ray bisects the angle. A crease merely passing through the vertex does not guarantee this: the ray-coincidence condition is essential. Similarly, folding two distinct points onto one another constructs their perpendicular bisector, since every crease point is equally distant from the two reflected points.

These proofs use established rigid-motion congruence facts. They do not assume a visually symmetric drawing establishes equal lengths. An executable construction identifies a circle’s center and radius, a line’s two determining points, and a choice among multiple intersections.

Apply

Write instructions another learner can execute without seeing your drawing. Exchange instructions and diagnose any unstated “choose the right point” step. For a copied angle, check both the radius and chord transfer; matching only a chord is insufficient. For a bisector, state the interior-ray choice and the nonstraight-angle hypothesis. If using paper, report observed discrepancies separately from the ideal correctness proof.

Continue

Use equal-distance constructions next to build perpendiculars and parallels through given points. These operations will become tools for inscribed figures, triangle centers and tangent constructions rather than isolated drawing tricks.