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Understanding Angle Relationships in Circles: Theorems and Applications

Explore the relationships between angles and arcs within circles through key theorems. This lesson covers the intersection of tangents and chords, along with their implications on angle measurements. Learn about the various theorems (10.11, 10.12, and 10.13) that govern angles inside and outside circles, and apply these concepts to find unknown angle measures. Engage with examples and practice problems that reinforce your understanding of angle-chord relationships and their significance in geometry.

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Understanding Angle Relationships in Circles: Theorems and Applications

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  1. Clickers Bellwork • One-half of the measure of an angle plus its supplement is equal to the measure of the angle. Find the measure of the angle • Solve for x • 2x2-13x-7=0

  2. Bellwork Solution • One-half of the measure of an angle plus its supplement is equal to the measure of the angle. Find the measure of the angle • Solve for x • 2x2-13x-7=0

  3. Section 10.5 Apply Other Angles Relationships in Circles

  4. The Concept • For the bulk of this chapter we’ve been dealing with the arc of circles • Today we’re going to finish out our discussion of arcs and angles

  5. Theorem Theorem 10.11 If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc C This makes sense if we think about this in terms of an inscribed angle with only one leg D A B

  6. On your own • What is the measure of arc ADC A D 82o C B

  7. On your own • What is the measure of arc ABC A D 82o C B

  8. On your own • What is the measure of angle ACB, if arc CD measures 160o D A 35o C B

  9. Theorems Theorem 10.12 If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle A C x B D

  10. Example Find the measure of angle AEC 110o A C E B 96o D

  11. On your own • What is the measure of angle ABC D 225o A B E 255o C

  12. On your own • What is the measure of angle ABD D 148o B A E 30o C

  13. On your own • What is the measure of arc AC D A 121o 142o B E C

  14. Theorems Theorem 10.13 Angles Outside the Circle Theorem If a tangent and a secant, two tangents or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs C A B D

  15. Theorems Find the measure of angle ADB A C D 250o 110o B

  16. On your own B • What is the measure of angle ABD D A 30o E 86o C

  17. On your own • What is the measure of angle ADB A 195o 65o D B

  18. On your own • What is the measure of arc BD A 164o D 32o B

  19. On your own • What is the measure of arc AB A 120o D 22o B

  20. On your own • What is the measure of angle of E, if arc ABC is 200o A D B C E

  21. On your own A • What is the measure of angle of F? 110o B 80o D C F

  22. On your own A 110o • What is the measure of angle of F? 105o B D 35o C F

  23. Homework • 10.5 • 1-20, 22-25

  24. Most Important Points • Angle-Chord Relationships that intercept inside a circle • Angle relationship that intersects outside the circle

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