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Lesson 45 – Angles and Arcs

Lesson 45 – Angles and Arcs. Homework Review. “Sheet 10 – Statements and Converses” ANY QUESTIONS???????????????. Problem of the Day. State the converse of the following. If you think the converse is true, rewrite the statement using iff . If you think it is false, give a counter example.

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Lesson 45 – Angles and Arcs

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  1. Lesson 45 – Angles and Arcs

  2. Homework Review • “Sheet 10 – Statements and Converses” • ANY QUESTIONS???????????????

  3. Problem of the Day • State the converse of the following. If you think the converse is true, rewrite the statement using iff. If you think it is false, give a counter example. • If a triangle has two congruent angles, then it is isosceles. • If a quadrilateral is a square, then it has four equal sides.

  4. Problem of the Day • Given chords XY, XZ and YZ are equidistant from the centre, C, of a circle, prove that the triangle XYZ is equilateral. • Given two circle with centre C (concentric circles) have chords AB and PQ respectively, CX is perpendicular to AB, and A-P-X-Q-B, prove that AP=BQ. • Given circles with centre A and B intersect at P and Q, and M is the midpoint of PQ, prove that A-M-B.

  5. Arcs, Angles and Chord Vocabulary

  6. C  B  A   D Arc – A Portion of a Circle’s Circumference • Minor Arc – An arc that is less than 180˚ • E.g. • Major Arc – An arc that is greater than 180˚ • E.g (specified using the endpoints and an internal point of the arc)

  7. C A B Central Angle – The Angle at the Centre of a Circle. • A central angle and a its intercepted arc are congruent. • They are both measured in degrees. • ACB is central angle subtended by arc AB • The measure of arc AB is equal to the measure of ACB • arcAB= ACB

  8. D C A B Inscribed Angle – The Angle Formed by Two Chords that Meet at the Same Point on a Circumference. • The measure of an inscribed angle is equal to half the measure of the arc intercepted by the inscribed angle. • The measure of an inscribed angle is equal to half the measure of a central angle subtended by the same arc. • E.g. If then AND arcAB=80˚

  9. D E F A B More Inscribed Angles: The Bow-Tie Rule • All inscribed angles subtended by same chord (AB) are equal. • E.g.

  10. C  Inscribed Angles and the Diameter • Inscribed angles subtended by the diameter measure 90˚. • The central angle would be 180˚ since the diameter is a straight line. • Inscribed angles are always equal to half of the central angle.

  11. X W Y Z Cyclic Quadrilateral – A Quadrilateral Inscribed in a Circle • Sum of the opposite angles of a cyclic quadrilateral are 180˚.

  12. C A B Sector – Part of the Interior of a Circle Bounded by Two Radii and the Arc Between them.

  13. minorsegment majorsegment Segment – Part of a Circle’s Interior Bounded by a Chord and Arc • Major segments are more than half of a circle’s area. • Minor segments are less than half of a circle’s area.

  14. You Try! • Complete “Sheet 11.1 [and] 11.2 – Angles and Arcs”

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