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All Circles are Similar

All Circles are Similar. A circle is a set of all points that are equidistant to a given center point. If you dilate a circle, it will never change shape. ALL CIRCLES ARE SIMILAR TO EACH OTHER. Chords, Secants, and Tangents. Chord: A line segment that begins and ends on the circle. CHORD.

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All Circles are Similar

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  1. All Circles are Similar • A circle is a set of all points that are equidistant to a given center point. • If you dilate a circle, it will never change shape. • ALL CIRCLES ARE SIMILAR TO EACH OTHER

  2. Chords, Secants, and Tangents Chord: A line segment that begins and ends on the circle. CHORD Draw and label the following lines on the circle in your notes.

  3. Chords, Secants, and Tangents Secant: A line that touches a circle at two points. SECANT

  4. Chords, Secants, and Tangents Tangent: A line, segment, or ray that touches a circle at exactly one point. TANGENT

  5. Chords, Secants, and Tangents Radius: A segment whose endpoints are the center and any point on the circle. RADIUS

  6. Chords, Secants, and Tangents Diameter: A chord that contains the center of the circle. DIAMETER

  7. Central Angle: An angle whose vertex is the center of the circle. is the minor arc is the major arc CENTRAL ANGLE Draw and label the following angles on the circle. Degree measure of a minor arc: Defined as the same as the measure of its corresponding central angle.

  8. Draw a central angle. Measure the angle with a protractor and label the angle. R 110° The central angle defines the arc. 110° A S

  9. This is also an intercepted arc. These are NOT Inscribed angles

  10. Draw an inscribed angle with the same intercepts as the central angle. Measure that angle and label it. R 110° 110° A 55° B S

  11. Draw another inscribed angle with the same intercepts. Measure that angle and label it. R 110° C 55° 110° A 55° B S

  12. Draw another inscribed angle with the same intercepts. Measure that angle and label it. R 110° C 55° 110° A 55° B S 55° D What is the relationship between all inscribed angles with the same intercepts? What is the relationship between the inscribed angle and its intercepted arc?

  13. Theorem 70: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc. R 110° 110° 55° B S

  14. R Theorem 71: If two inscribed angles of a circle intercept the same arc or arcs of equal measure, then the inscribed angles have equal measure. C 55° 55° B S 55° D

  15. Inscribed Angle and Intercepted Arc Notice that the 72 degree angle and the red angle have the same intercepted arc. This means that angle Y and angle X are the same measure.

  16. Can you tell which angle is congruent to angle U? Where is their intercepted arc?

  17. Draw a diameter of the circle. Draw an inscribed angle with intercepts on the endpoints of the diameter. A Can you explain why is a right angle? O 90° C B

  18. Theorem 72: If an inscribed angle intercepts a semicircle, then its measure is 90°. A O 90° C B

  19. Draw four cords in this circle to create a quadrilateral. Measure and label all four angles 78° What is the relationship between opposite angles in the quadrilateral? Think of the inscribed angle measures compared to the central angles and the total degrees in a circle. 107° 73° 102°

  20. Inscribed Polygons A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

  21. Find the value of each variable.

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