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Understanding Circle Angle Relationships: Tangents, Chords, and Secants

This guide covers the relationships between angles formed by tangents, chords, and secants within and around circles. Theorems include: (1) Tangent and chord intersection on a circle (Theorem 10.12), where each angle is half the intercepted arc; (2) Chord intersection inside the circle (Theorem 10.13), where angles equal half the sum of intercepted arcs; (3) Intersections outside the circle (Theorem 10.14), where angles are half the difference of intercepted arcs. Includes practice problems for better understanding.

uriel-middleton
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Understanding Circle Angle Relationships: Tangents, Chords, and Secants

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  1. Section 10.4 Other Angle Relationships in Circles

  2. Intersection ON the circle(theorem 10.12) • If a tangent and a chord intersect at a point on the circle, then the measure of each angle formed is one half the measure of the intercepted arc Ex: m1 = ½ m AB m2 = ½ m BCA 120° m1 = ½ (120 °)=60°

  3. Intersection INSIDE the circle(theorem 10.13) • If two chords intersect inside the circle, then each angle measures one half the sum of the measures of the intercepted arcs m1 = ½ (m CD + m AB) Ex: m1 = ½ (100° + 30°) = 65° 30° 100°

  4. Intersection OUTSIDE the circle(theorem 10.14) • If a tangent and a secant, two tangents, or two secants intersect outside the circle, then each angle measures one half the difference of the intercepted arcs.

  5. Tangent and Secant B m1 = ½ (m BC – m AC) A 200° 60° 1 D C m1 = ½ (200° - 60 °)=70°

  6. Two Tangents m2 = ½ (m PQR – m PR) P 280° 80° 2 Q R m2 = ½ (280° - 80 °)=100°

  7. Two Secants m3 = ½ (m XY – m WZ) X W 30° 3 100° Z Y m3 = ½ (100° - 30 °)=35°

  8. Practice Problems • Intersection on the circle: Angle = ½ arc • Intersection inside the circle: Angle = (arc + arc) 2 • Intersection outside the circle: Angle = large arc – small arc 2

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