1. CIRCLES 2
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2. ANGLE PROPERTIES:
Let’s review the methods for finding the arcs and the different kinds of angles found in circles.
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3. The measure of a minor arc is the same as…
…the measure of its central angle.
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4. Example:
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5. The measure of an inscribed angle is…
…half the measure of its intercepted angle.
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6. Example:
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7. The measure of an angle formed by a tangent and secant is …
…half the measure of its intercepted arc.
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8. Example:
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9. ...is half the sum of the two intercepted arcs.
The measure of one of the vertical angles formed by 2 intersecting chords
...is half the sum of the two intercepted arcs.
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10. Example:
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11. …half the difference of the measures of its two intercepted arcs.
The measure of an angle formed by 2 secants intersecting outside of a circle is…
…half the difference of the measures of its two intercepted arcs.
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12. Example:
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13. …half the difference of the measures of its two intercepted arcs.
The measure of an angle formed by 2 tangents intersecting outside of a circle is…
…half the difference of the measures of its two intercepted arcs.
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14. Example:
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15. PROPERTIES: Complete the theorem relating the objects pictured in each frame.
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16. Note: Many of our theorems begin the same way, “In the same circle, or in congruent circles…”
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17. So: We will just start “In the same circle. …” where the
So: We will just start “In the same circle*…” where the * represents the rest of the phrase.
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18. All radii in the same circle,* …
...are congruent.
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19. In the same circle,* Congruent central angles...
...intercept congruent arcs.
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20. In the same circle,* Congruent Chords...
...intercept congruent arcs.
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21. Tangent segments from an exterior point to a circle…
...are congruent.
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22. The radius drawn to a tangent at the point of tangency…
...is perpendicular to the tangent.
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23. If a diameter (or radius) is perpendicular to a chord, then…
...it bisects the chord…
…and the arcs.
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24. In the same circle,* Congruent Chords...
...are equidistant from the center.
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25. Example: Given a circle of radius 5” and two 8” chords
Example: Given a circle of radius 5” and two 8” chords. Find their distance to the center.
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26. If two Inscribed angles intercept the same arc...
...then they are congruent.
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27. If an inscribed angle intercepts or is inscribed in a semicircle …
...then it is a right angle.
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28. If a quadrilateral is inscribed in a circle then each pair of opposite angles …
...must be supplementary.
(total 180o)
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29. If 2 chords intersect in a circle, the lengths of segments formed have the following relationship:
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30. Example:
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31. If 2 secants intersect outside of a circle, their lengths are related by…
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32. Example:
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33. If a secant and tangent intersect outside of a circle, their lengths are related by…
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34. Example:
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35. Let’s Practice!

36. Example: Given
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37. Example:
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38. Example:
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39. Example:
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40. Example: Given a circle of radius 13” and two 24” chords
Example: Given a circle of radius 13” and two 24” chords. Find their distance to the center.
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41. Example:
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42. Example:
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43. Example:
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44. Example:
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45. Example:
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46. Example:
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47. Example: Of the following quadrilaterals, which can not always be inscribed in a circle?
Rectangle
Rhombus
Square
Isosceles Trapezoid

48. Example:
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49. Example:
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50. Example: Regular Hexagon ABCDEF is inscribed in a circle.
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51. THE END!
Now go practice!