CIRCLES AND TANGENTS THEOREMS

CIRCLES AND TANGENTS THEOREMS

1. Theorem 1

   • If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

2. Theorem 2

   • If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.
   • Corollary 1
     • Two tangent segments from a common external point are congruent.
     • Corolary 2
       • The two tangent rays from a common external point determine an angle that is bisected by the ray from the external point to the center of the circle.

3. Theorem 5 - The Arc Addition Theorem

   • The measure of an arc formed by two adjacent nonoverlapping arcs (Arcs that share exactly one point) is equal to the sum of the measures of these two arcs.
     • Arc AC = arc AB + arc BC
     • Inscribed Angle
       • Is an angle formed by the intersection of two chords at a point in the circle.

4. Theorem 6

   • The measure of an inscribed angle is equal to one-half the measure of its intercepted arc.
   • Corollary 3
     • An angle inscribed in a semicircle is a right angle.
     • Corollary 4
       • Angles inscribed in the same arc are congruent.
       • Corollary 5
         • The opposite angles of an inscribed quadrilateral are supplementary.

5. Theorem 7 – The Tangent-Secant Theorem

   • Given an angle with its vertex on a circle, formed by a secant ray and a tangent ray, the measure of the angle is half the measure of the intercepted arc.

6. Theorem 8

   • If two chords intersect within a circle, then the measure of the angle formed is equal to half the sum of the measures of the intercepted arc.

7. Theorem 9

   • If a tangent and a secant, two secants, or two tangents intersect in a point in the exterior of a circle, then the exterior of a circle, then the measure of the angle formed is equal to one-half the difference of the measures of the intersected arcs.

8. Theorem 10

   • If two chords intersect inside a circle, then the product of the lengths of the segment parts of one chord is equal to the product of the lengths of the segment parts of the other chord.

9. Theorem 11

   • If two secants intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment part is equal to the product of the lengths of the other segment and its external segment part.

10. Theorem 12

    • If a secant and a tangent intersect in the exterior of a circle, the product of the lengths of the secant segment and its external segment part is equal to the square of the length of the tangent segment.