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Circles!

This comprehensive guide covers essential vocabulary and concepts related to circles, including major and minor arcs, tangents, secants, chords, central and inscribed angles. It introduces the concept of the Magic Number to calculate various angle measures, highlighting relationships between angles and arcs. The document also explores the properties of right angles and tangents and provides proofs involving kites and congruent triangles. Perfect for geometry students looking to enhance their understanding of circles and angle relationships.

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Circles!

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  1. Circles! Arcs & Angles, Measurement

  2. Arcs & Angles Circle Vocabulary Major Arc: Arc greater than 180⁰. Minor Arc: Arc less than 180⁰. Tangent: Line that intersects the circle at 1 point. Secant: Line that intersects the circle at 2 point. Chord: Segment whose endpoints are on the circle. Central Angle: Angle whose vertex is at the center. Inscribed Angle: Angle whose vertex is on the circle. DBA DA D AE DB C A DA E <BCA B <BDA

  3. Arcs & Angles Find the Magic Number! m<1 + m<2 + m<3 = ? D 1 mDA + mAB + mBD = ? C 2 A Sum of the Angles = ? Sum of the Arcs 3 B

  4. Outside Angle C 1 Use the Magic Number! ½ (big arc – small arc) = angle D B D C A E A B A 1 1 B C D

  5. Inside Angle Use the Magic Number! ½ arc = inscribed angle ½ (arc + arc) = angle A D B 1 C 1 B C A D

  6. Angles Both Outside & Inside Use the Magic Number! D ½ mDBA = m<2 1 C A ½ mDA = m<1 2 E B

  7. Right Angles & Circles mDAB = ? 180⁰ E m<1 = ? 90⁰ D ? radius. DC is a 1 ? tangent. DE is a A C The tangent is perpendicular to the radius. B

  8. Right Angles & Circles Prove DE ~ EA EACD is a kite! E Given ED , EA are tangent to C at D, A respectively. 1 If ED , EA are tangent, then <EDC and <EAC are right angles. D EC~ EC by the reflexive property. DC~ AC because they are radii. A ΔEDC ~ ΔEAC by HL (hyp-leg). C DE ~ EA by CPCTC B

  9. Angles & Circles Prove <1 ~ <2 Given DB ~ EA Remember <1 & <2 are NOT vertical! E D C 2 1 A B

  10. Angles & Circles Prove <1 ~ <2 Given DB ~ EA Remember <1 & <2 are NOT vertical! E D C 2 1 A B

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