The difference of the angles, or F A B, must be measured by half the difference of the arcs, or half of F B. [Illustration] PROPOSITION XXI. THEOREM. DEMONSTRATION. We wish to prove that _Parallel chords intercept equal arcs of the circumference._ Let the chords A B, C D, be parallel; then will the intercepted arcs A C and B D be equal. For draw the straight line B C. Because the lines A B and C D are parallel, the interior alternate angles A B C, B C D, are equal. But the angle A B C is measured by half the arc A C; And the angle B C D is measured by half the arc B D: Then, because the angles are equal, the half arcs which measure them must be equal, and the whole arcs themselves must be equal. [Illustration] PROPOSITION XXII. THEOREM. DEMONSTRATION. We wish to prove that _The angle formed by a tangent and a chord meeting at the point of contact is measured by half the intercepted arc._ Let the tangent C A B and the chord A D meet at the point of contact A; then will the angle B A D be measured by half the intercepted arc A D. For draw the diameter A E F. Because A B is a tangent, and A E a radius at the point of contact, the angle B A F is a right angle, and is measured by the semicircle A D F. Because the angle F A D is at the circumference, it is measured by half the arc D F. Then the difference between the angles B A F and D A F, or B A D, must be measured by half the difference of the arcs A D F and D F, or A D; That is, the angle B A D is measured by half the arc A D. [Illustration] PROPOSITION XXIII. THEOREM. DEMONSTRATION. We wish to prove that _A tangent and chord parallel to it intercept equal arcs of the circumference._ Let A B be tangent to the circumference at the point D, and let C F be a chord parallel to the tangent; then will the intercepted arcs C D and D F be equal. For from the point of contact D, draw the straight line D C. Because the tangent and chord are parallel, the interior alternate angles A D C and D C F are equal. But the angle A D C, being formed by the tangent D A and the chord D C, is measured by half the intercepted arc D C; And the angle D C F, being at the circumference, is measured by half the arc on which it stands, D F: Then, because the angles are equal, the half arcs which measure them are equal, and the arcs themselves are equal. [Illustration] PROPOSITION XXIV. THEOREM. DEMONSTRATION. We wish to prove that _The angle formed by the intersection of two chords in a circle is measured by half the sum of the intercepted arcs._