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Chapter 10 - Circles Objectives: Use arcs, angles, & segments in circles to solve problems Use the graph of an equation of...
10.2 & 10.3 Arcs & Chords & Inscribed Angles Objectives: Use properties of arcs of circles. Use properties of chords of ci...
Arc terms <ul><li>In a plane, an angle whose vertex is the center of a circle is a  central angle  of the circle. </li></u...
Arc terms <ul><li>Arcs are named by their endpoints.  The minor arc in this picture is Arc GF. </li></ul><ul><li>Major arc...
Arc terms <ul><li>The measure of a minor arc is defined to be the measure of its central angle. </li></ul><ul><li>The meas...
Example <ul><li>What is the measure of Arc MP? </li></ul><ul><li>80 </li></ul><ul><li>What is the measure of Arc MNP? </li...
Arc Addition Postulate <ul><li>The measure of an arc formed by 2 adjacent arcs is the sum of the measures of the 2 arcs. <...
Congruence <ul><li>Two arcs of he same circle or of congruent circles are  congruent arcs  if they have the same measure. ...
Theorems about Chords  (see pictures on p. 605 & 606) <ul><li>In the same circle, or in congruent circles, 2 minor arcs ar...
Do Masonry Hammer example <ul><li>On p. 606.  </li></ul>
Inscribed Angle <ul><li>An  inscribed angle  is an angle whose vertex is on a circle and whose sides contain chords of the...
Measure of an Inscribed Angle <ul><li>If an angle is inscribed in a circle, then its measure is half the measure of the in...
Another Theorem <ul><li>If two inscribed angles of a circle intercept the same arc, then the angles are congruent. </li></...
Do Example 4, p. 614
Inscribed polygons <ul><li>If all the vertices of a polygon lie on a circle, the polygon is  inscribed  in the circle </li...
Theorems about inscribed polygons <ul><li>If a right triangle is inscribed in a circle, then the hypotenuse is a diameter ...
Theorems about inscribed polygons <ul><li>A quadrilateral can be inscribed in a circle if and only if its opposite angles ...
Do example 6, p. 616 <ul><li>Do worksheets </li></ul>
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Geom10point2and3.Doc

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  1. 1. Chapter 10 - Circles Objectives: Use arcs, angles, & segments in circles to solve problems Use the graph of an equation of a circle to model problems
  2. 2. 10.2 & 10.3 Arcs & Chords & Inscribed Angles Objectives: Use properties of arcs of circles. Use properties of chords of circles Use properties of inscribed angles Understand inscribed polygons
  3. 3. Arc terms <ul><li>In a plane, an angle whose vertex is the center of a circle is a central angle of the circle. </li></ul><ul><li>If the measure of a central angle is less than 180˚, then that angle is a minor arc of the circle. </li></ul><ul><li>If the measure of a central angle is greater than 180˚, then that angle is a major arc of the circle. </li></ul><ul><li>If the endpoints of an arc are the </li></ul><ul><li>endpoints of a diameter, then the </li></ul><ul><li>arc is a semicircle . </li></ul>Minor arc Major arc
  4. 4. Arc terms <ul><li>Arcs are named by their endpoints. The minor arc in this picture is Arc GF. </li></ul><ul><li>Major arcs and semicircles are named by their endpoints and a point on the arc. </li></ul><ul><li>The major arc associated with GHF is Arc GEF. </li></ul><ul><li>Arc EGF is a semicircle. </li></ul>G H F 60˚ E
  5. 5. Arc terms <ul><li>The measure of a minor arc is defined to be the measure of its central angle. </li></ul><ul><li>The measure of Arc GF is 60˚. </li></ul><ul><li>The measure of a major arc is defined as the difference between 360˚ and the measure of its associated minor arc. </li></ul><ul><li>The measure of Arc GEF is 300˚. </li></ul>G H F 60˚ E
  6. 6. Example <ul><li>What is the measure of Arc MP? </li></ul><ul><li>80 </li></ul><ul><li>What is the measure of Arc MNP? </li></ul><ul><li>280 </li></ul><ul><li>What is the measure of Arc PMN? </li></ul><ul><li>180 </li></ul>M R P 80˚ N
  7. 7. Arc Addition Postulate <ul><li>The measure of an arc formed by 2 adjacent arcs is the sum of the measures of the 2 arcs. </li></ul>
  8. 8. Congruence <ul><li>Two arcs of he same circle or of congruent circles are congruent arcs if they have the same measure. </li></ul><ul><li>Are these 2 arcs congruent? </li></ul>
  9. 9. Theorems about Chords (see pictures on p. 605 & 606) <ul><li>In the same circle, or in congruent circles, 2 minor arcs are congruent if and only if their corresponding chords are congruent. </li></ul><ul><li>If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. </li></ul><ul><li>If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. </li></ul><ul><li>In the same circle, or in congruent circles, 2 chords are congruent if and only if they are equidistant from the center. </li></ul>
  10. 10. Do Masonry Hammer example <ul><li>On p. 606. </li></ul>
  11. 11. Inscribed Angle <ul><li>An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. </li></ul><ul><li>The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. </li></ul>
  12. 12. Measure of an Inscribed Angle <ul><li>If an angle is inscribed in a circle, then its measure is half the measure of the intercepted arc. </li></ul>A B
  13. 13. Another Theorem <ul><li>If two inscribed angles of a circle intercept the same arc, then the angles are congruent. </li></ul>A B
  14. 14. Do Example 4, p. 614
  15. 15. Inscribed polygons <ul><li>If all the vertices of a polygon lie on a circle, the polygon is inscribed in the circle </li></ul><ul><li>And the circle is circumscribed about the polygon </li></ul>
  16. 16. Theorems about inscribed polygons <ul><li>If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. </li></ul><ul><li>Conversely, if 1 side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. </li></ul>
  17. 17. Theorems about inscribed polygons <ul><li>A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. </li></ul>
  18. 18. Do example 6, p. 616 <ul><li>Do worksheets </li></ul>

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