Question

What is the construction that needs to be done to make a copy of ∠BAC ?

Answer 1

To copy angle ∠BAC at a new vertex D (or onto a given ray from D), use this straightedge-and-compass construction:

Steps
1. With center A draw any arc that meets ray AB at E and ray AC at F.
2. With the same compass width, draw an arc centered at D. If you already have a starting ray from D, let this arc meet that ray at G; otherwise choose any ray from D and let the arc meet it at G.
3. Measure the distance EF with the compass (leave the compass at that opening).
4. With center G and radius EF draw an arc that meets the arc centered at D at H.
5. Draw the ray from D through H. The angle between the initial ray (DG) and this new ray (DH) is congruent to ∠BAC.

Why it works (brief)
- The arcs AE and AF are equal radii, so points E and F are equally spaced from A; EF encodes the opening of ∠BAC.
- Reproducing the same arc at D gives points G and H at the same radius from D, and setting GH = EF recreates the same subtended chord length. By the congruence of the corresponding isosceles triangles, the angle at D equals ∠BAC.

If you want the construction placed on a specific given ray from D, use that ray as the initial ray in step 2.