Other angle relationships
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Other Angle Relationships. Section 10.6. Tangent-Chord Theorem. If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. . Example 1. Try This!. Example 2. Interior Intersection Theorem.

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Tangent chord theorem
Tangent-Chord Theorem

  • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.





Interior intersection theorem
Interior Intersection Theorem

  • If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.


Exterior intersection theorem
Exterior Intersection Theorem

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


Diagrams for exterior intersection theorem
Diagrams for Exterior Intersection Theorem


Example 3
Example 3

  • Find the value of x.


Try this1
Try This!

  • Find the value of x.


Example 4
Example 4

  • Find the value of x.


Example 5
Example 5

  • Find the value of x.


Chord product theorem
Chord Product Theorem

  • If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.


Example 11
Example 1

  • Find the value of x.


Try this2
Try This!

  • Find the value of x.


Secant secant theorem
Secant-Secant Theorem

  • If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.


Secant tangent theorem
Secant-Tangent Theorem

  • If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.


Example 21
Example 2

  • Find the value of x.


Try this3
Try This!

  • Find the value of x.


Example 31
Example 3

  • Find the value of x.


Try this4
Try This!

  • Find the value of x.


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