Angles between chords, secants and a tangent
An angle whose vertex is not on the circle is measured by arcs: between two crossing chords, half the sum of the arcs; between two secants, half the difference; between a tangent and a chord, half the arc.
- Russia: grade 8
Guide
In the Russian programme the inscribed angle is Grade 8 geometry, chapter "Circle": it is half the arc it stands on. Angles between chords, secants and a tangent follow from it and appear in harder problems.
Three facts. The angle between two chords crossing inside the circle is half the sum of the arcs inside that angle and inside its vertical angle. The angle between two secants from an outside point is half the difference of the larger and smaller arcs. The angle between a tangent and a chord through the point of contact is half the arc inside the angle.
Move the points and compare the angle with the arcs (the central angles). Lay solutions out as Given, Find, Solution, Answer; figures are not to scale.
The "why" question asks for your own explanation; then compare it with the model.