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Understanding Circle Geometry: Inscribed Angles and Theorems

Dive into the fundamental concepts of circle geometry, focusing on essential elements such as radius, diameter, circumference, and pi. Explore the specifics of inscribed angles, including the Inscribed Angle Theorem, detailing how the measure of an inscribed angle is half that of its intercepted arc. Learn through examples, corollaries, and proofs, including the properties of inscribed quadrilaterals and the relationships between tangent and radius. This recap is a vital resource for mastering circle-related concepts.

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Understanding Circle Geometry: Inscribed Angles and Theorems

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  1. Day 70:Circles The Return

  2. Recap • Radius • Diameter • Circumference • Pi • Secant • Tangent • Chord • Area • Central Angle • Arc • Arc Measure • Arc Length • Sector • Area of a Sector

  3. Inscribed Angles • An inscribed angle has its vertex on the circle, and its rays intersect the circle • ACB is an inscribedangle. • The arc formed by aninscribed angle is calledan intercepted arc. • AB is an intercepted arc

  4. Inscribed Angle Theorem • The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. • mB = ½(arc AC) • arc AC = 2(mB) • mC = ½(arc AB) • arc AB = 2(mC) • mA = ½(arc BC) • arc BC = 2(mA)

  5. Inscribed Angle Theorem • Example: • If mADB = 24, whatis the measure of AB? • What is mACB? • If AB = 36, whatis mADB? • What is mACB?

  6. Inscribed Angle Theorem Proof • Given: B inscribed inסּO • Prove: mB = ½ AC • First we construct radius OC. • This makes isosceles triangleBOC. • mAOC = mB + mC (exteriorangle) • Since mB = mC, then mAOC = 2mB. • B = ½ mAOC = ½ AC B O C A

  7. Inscribed Angle Theorem Corollary • If two angles inscribe the same arc, or congruent arcs, then the angles are congruent. • Example: B and D bothinscribe AC. • Therefore, B  D B D O C A

  8. Inscribed Angle Theorem Corollary • We previously demonstrated that the arc measure of a semicircle, such as PRQ, is 180. • What is the measureof an angle inscribed bya semicircle (e.g. PRQ)? • An inscribed angle of atriangle intercepts adiameter or semicircle if andonly if it is a right triangle.

  9. Inscribed Quadrilateral Theorem • A quadrilateral can be inscribed inside of a circle if and only if its opposite angles are supplementary. • Example: mA + mC = 180and mB + mD = 180 • Proof: B and D inscribeadjacent arcs that create thewhole circle. Since the two arcsadd to make 360, and the angles are half of the arcs, then the two angles add to 180 C B O D A

  10. Tangent/Radius Theorem • Any tangent of a circle is perpendicular to a radius of that circle at their point of intersection. • Indirect Proof • If a diameter intersectsa tangent at each end,those tangents areparallel.

  11. Intersecting Tangents • If two segments from the same exterior point are tangent to a circle, then those segments are congruent. • AP and AQ are tangent tothe circle. • AP  AQ. P Q A

  12. Homework 43 • Workbook, pp. 129, 131

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