[hyperpallium]

Geometry distinguishes between equivalence and value. An "angle" isn't its degrees, but the geometric figure (two rays or segments meeting at an end-point of each). It's the measure of the angle that is the degrees.

You don't say "angles are equal" - you say they are congruent. It's their measures that are "equal".

Although congruency implies measure equality, it doesn't really mean that, but that the shapes are the same (can be rotated/translated to coincide).

[fjsolwmv]

Congruent means that you have some relevant information that you are discarding (projecting away) information, such as position and rotation. It's just as correct to say equal if you've already established the relevant context / quotient space.

Two triangles on a page are congruent, because they have different position. The three corners of an eauikao trisne are congruent, because their corner+angles are equal. If you take angle to mean corner, you say congruent. If you aren't also talking so about their position, you say equal.

In math you can slice things every which way, so it's impossible to use the different words for every different concept, so you have to establish a context

Equal vs congruent (and equal vs equivalent, which are also synonymous in math) is a crutch for beginners who are over reliant on their informal intuition.

[vinchuco]

Measures are equal if and only if angles are congruent ?