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Bell work 1

Bell work 1. Find the measure of the inscribed angles , R, given that their common intercepted TU = 92º. T. •. TU = 92 º. R. •. •. U. Bell work 1 Answer. Angles R = ½ the intercepted arc TU since their intercepted Arc TU = 92º, then Angle R = 46º. T. •. TU = 92 º. R. •. •.

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Bell work 1

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  1. Bell work 1 Find the measure of the inscribed angles , R, given that their common intercepted TU = 92º T • TU = 92º R • • U

  2. Bell work 1 Answer Angles R = ½ the intercepted arc TU since their intercepted Arc TU = 92º, then Angle R = 46º T • TU = 92º R • • U

  3. Bell work 2 A quadrilateral WXYZ is inscribed in circle P, if ∕_ X = 130º and ∕_ Y = 106º , Find the measures of ∕_ W = ? and ∕_ Z = ? The Quadrilateral WXYZ is inscribed in the circle iff / X + / Z = 180º, and / W + / Y = 180º X Y • • • 130º 106º P • W • Z

  4. Bellwork 2 Answer From Theorem 10.11 ∕_ W = 180º – 106º = 74º and ∕_ Z = 180º – 130º = 50º The Quadrilteral WXYZ is inscribed in the circle iff / X + / Z = 180º, and / W + / Y = 180º X Y • • • 130º 106º P • W • Z

  5. Unit 3 : Circles: 10.4 Other Angle Relationships in Circles Objectives: Students will: 1. Use angles formed by tangents and chords to solve problems related to circles 2. Use angles formed by lines intersecting on the interior or exterior of a circle to solve problems related to circles

  6. (p. 621) Theorem 10.12 If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is ½ the measure of its intercepted arc A • m∕_ 1 = ½ m minor AC P m∕_ 2 = ½ m Major ABC B Angle 1 • • Angle 2 • C m

  7. (p. 621) Theorem 10.12 Example 1 Find the measure of Angle 1 and Angle 2, if the measure of the minor Arc AC is 130º A • m minor AC = 130º P B • • Angle 1 Angle 2 • C m

  8. (p. 621) Theorem 10.12 Example 1 Answer The measure of Angle 1 = 65º and Angle 2 = 115º A • m minor AC = 130º P B • • 65º = Angle 1 Angle 2 = 115º • C m

  9. (p. 621) Theorem 10.12 Example 2 Find the measure of Angle 1, if Angle 1 = 6xº, and the measure of the minor Arc AC is (10x + 16)º m minor AC = (10x + 16)º A • Angle 1= 6xº P • • C • B m

  10. (p. 621) Theorem 10.12 Example 2 Answer Angle 1 = 6xº = ½ Arc AC = ½ (10x + 16)º 6xº = ½ (10x + 16)º 6xº = 5x + 8 x = 8º thus, Angle 1 = 48º A • m minor AC = (10x + 16)º P • • Angle 1= 6xº C • B m

  11. Intersections of lines with respect to a circle There are three places two lines can intersect with respect to a circle. • • • Outside the cirlce On the circle In the circle

  12. (p. 622) Theorem 10.13 If two chords intersect in the interior of a circle , then the measure of each angle is ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle. D Angle 1 m∕_ 1 = ½ (m AB + m CD) • C • P • Angle 2 • A m∕_ 2 = ½ (mBC + mAD) • B

  13. (p. 622) Theorem 10.13Example Find the value of x. m CD = 16º D • C • P • Angle 1 xº • A • m AB = 40º B

  14. (p. 622) Theorem 10.13Example Answer x = ½ (m AB + m CD) = ½ (40º + 16º) x = ½ (56º) x = 28 º m CD = 16º D • C • P • Angle 1 xº • A • m AB = 40º B

  15. (p. 622) Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is ½ the difference of the intercepted arcs. 1 Tangent and 1 Secant 2 Tangents 2 Secants P X B • • • W A • • • 2 • • 3 1 • • Q • Z • • R Y C • m∕_ 1 = ½ (m BC – m AC) m∕_ 2 = ½ (m PQR – m PR) m∕_ 3 = ½ (m XY – m WZ) B

  16. (p. 622) Theorem 10.14Example 1 Find the value of x P • • Major Arc PQR = 266º xº • Q • R m∕_ x = ½ (m PQR - mPR)

  17. (p. 622) Theorem 10.14Example 1 Answer m PR = (360º - m PQR) = (360º - 266º) = 94º x = ½ (m PQR - m PR) = ½ (266º - 94º) = ½ (172º) x = 86 º P • • Major Arc PQR = 266º xº • Q • R m∕_ x = ½ (m PQR - mPR)

  18. (p. 622) Theorem 10.14Example 2 Find the value of x, GF. The m EDG = 210º The m angle EHG = 68º E • F Major Arc EDG = 210º • D • • xº 68º H • G m∕_ EHG = 68º = ½ (m EDG – m GF)

  19. (p. 622) Theorem 10.14Example 2 Answer m∕_ EHG = 68º = ½ (m EDG – m GF) 68º = ½ ( 210º - xº ) 136º = 210º - xº xº = 210º - 136º xº = 74º E • F Major Arc EDG = 210º • D • • xº 68º H • G m∕_ EHG = 68º = ½ (m EDG – m GF)

  20. Home work PWS 10.4 A P. 624 (8 -34) even

  21. Journal Write two things about “the intersections of chords, secants, and/or tangents” related to circles from this lesson.

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