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6.6 – Secants, Tangents, and Proofs

Key Terms

  • Point of Tangency – The point at which a tangent line meets a curve.
    • In a circle, the radius ending at the point of tangency is always perpendicular to the tangent line.
  • Secant – A line or a line segment that intersects a circle in two points.
  • Secant-Secant Angle – The angle formed by the intersection of two secants of the same circle.
    • The measure of a secant-secant angle is half the difference of the measures of the intercepted arcs.
  • Secant-Secant Theorem – The measure of a secant-secant angle is one-half of the difference between the two intercepted arcs.
  • Tangent Line – A line that intersects a circle at exactly one point, known as the point of tangency.
  • Tangent-Chord Angle – An angle formed by a tangent and a chord that shares the point of tangency.
    • The measure of a tangent-chord angle is half the measure of the intercepted arc inside the angle.
  • Tangent-Tangent Angle – An angle formed by intersecting tangents.
    • The measure of a tangent-tangent angle is half the difference of the measures of the intercepted arcs.

 


Review

  • m means “measure” in degrees
  • Inscribed angles are half the measure of their intercepted arcs

 


Notes

  • Secants

GeoB 6.6 Secant 01

GeoB 6.6 Secant

 

  • Secants vs Chords

GeoB 6.6 Secant Chord

 

  • Secant-Secant Angle
    • Formula: \angle ACE=\frac{1}{2}(m \widehat{AE}-m \widehat{BD})

GeoB 6.6 Secant Ex 01 GeoB 6.6 Secant Secant

 

  • Secant-Secant Theorem
    • Draw chord \overline{AD} because you can!
    • That chord forms two inscribed angles: 26° and 68°

GeoB 6.6 Secant Secant Triangle

 

  • Remote Exterior Angles of a Triangle and the Secant-Secant Angle Theorem

GeoB 6.6 Secant Secant Theorem

 

  • Watch the Video and Copy the Proof in the Template Below

Geometric Proof Template

 


  • Tangent
    • An important property of a tangent is that it is perpendicular to the radius that shares the point of tangency.

GeoB 6.6 Tangent ex GeoB 6.6 Tangent Img

 

  • Tangent vs Secant

GeoB 6.6 Tangent

 

  • Tangent-Chord Angle
    • Tangent-Chord Angle is that the measure of the angle is half the measure of the arc inside it.
    • The same concept as an inscribed angle and an intercepted arc.

GeoB 6.6 Tangent Chord

GeoB 6.6 Tangent Chord Rule

 

  • Tangent-Tangent Angle
    • Formula: \angle ABC=\frac{1}{2}(m \widehat{ADE}-m \widehat{AC})
    • The measure of a tangent-tangent angle is half the difference of the measures of the intercepted arcs

GeoB 6.6 Tangent Tangent Angle Ex

GeoB 6.6 Tangent Tangent Angle2

 

  • Tangent-Tangent Angle vs Secant-Secant Angle

GeoB 6.6 Tangent Tangent Secant Secant

 

  • Remote Exterior Angles of a Triangle and the Tangent-Tangent Angle Theorem

GeoB 6.6 Tangent Tangent Angle Triangle

 

  • Watch and Copy the Proof into the Template Below

Geometric Proof Template

 

  • Tangent-Tangent Angle Formula & Rule

GeoB 6.6 Tangent Tangent Angle Rule

 


Examples

  • Do the mandatory problems for some great examples!

 


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