Congruent Angle Construction Action!

The applet below illustrates how we can use a compass and straightedge to[b] construct an angle congruent to [/b][color=#ff00ff][b]a given angle[/b][/color]. [br][br][b][color=#0000ff]Can you formally prove why this method of construction is valid? [/color][/b][br](For a hint, [url=https://www.geogebra.org/m/fskReBGs]click here[/url].)

Information: Congruent Angle Construction Action!