Subtend

Many theorems in geometry relate to subtended angles. If two sides of a triangle are congruent, then the angles they subtend are also congruent, and conversely if two angles are congruent then they are subtended by congruent sides (propositions I.5–6 in Euclid's Elements), forming an isosceles triangle. More generally, the law of sines states that the sine of each angle of a triangle is proportional to the side subtending it. The inscribed angle theorem states that when the vertex of an angle inscribed in a circle lies on the same side of the chord subtending it as the center of the circle, then the central angle subtended by the same chord is twice the inscribed angle.