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

• Secants vs Chords

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

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

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

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

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

• Tangent vs Secant

• 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.

• 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

• Tangent-Tangent Angle vs Secant-Secant Angle

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

• Watch and Copy the Proof into the Template Below

• Tangent-Tangent Angle Formula & Rule

Examples

• Do the mandatory problems for some great examples!