Question

how do you know that each of these constructions is valid when it is made with a compass and straightedge?

Explanation:

Compass - and - straightedge constructions are based on Euclidean geometry principles. Valid constructions follow the rules of geometric axioms and postulates. For example, constructing an equilateral triangle using these tools relies on the fact that circles of equal radius centered at the endpoints of a line - segment will intersect to form equal - length sides, which is in line with the properties of circles and congruence in Euclidean geometry.

Answer:

These constructions are valid when they follow the rules and principles of Euclidean geometry such as congruence postulates, circle properties, and straight - line axioms.