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Warm Up Section 3.3 Find x. In each figure the segment is tangent to the circle.

Warm Up Section 3.3 Find x. In each figure the segment is tangent to the circle. 1. 2. 3. 4. 5. 12. 15. 5. 13. x. 60 o. x. 15. x. 2. 3 x + 2. 4 x – 7. x. 7. x.

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Warm Up Section 3.3 Find x. In each figure the segment is tangent to the circle.

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  1. Warm Up Section 3.3 Find x. In each figure the segment is tangent to the circle. 1. 2. 3. 4. 5. 12 15 5 13 x 60o x 15 x 2 3x + 2 4x – 7 x 7 x

  2. Answers to Warm Up Section 3.3 Find x. In each figure the segment is tangent to the circle. 1. 2. 3. 4. 5. x = 13 x = 30 x ≈ 7.5 12 15 5 13 x 60o x 15 x 2 3x + 2 4x – 7 x 7 x x = 11.25 x = 9

  3. 3.2 Homework Answers • 122° 2. 90° 3. 110 ° 4. 20 • 5. 80 ° 6. 180 ° 7. 160o 8. 80o • x = 1 10. x = 10 11. x = 9 12. x = 6 • R 14. • 15. 16. • 17. 18. 19. 94o 20. 86o • 21. 266o 22. 266o

  4. Inscribed Angles and Polygons Section 3.3 Standard: MCC9-12.G.C.2 Essential Question: Can I understand and use properties of inscribed angles to solve problems?

  5. Vocabulary: Central angle: an angle whose vertex is the center of a circle Inscribed angle: an angle whose vertex is on a circle and whose sides contain chords of the circle. Intercepted arc: the arc that lies in the interior of an inscribed angle and has endpoints on the angle. Inscribed polygon: a polygon in which all of its vertices lie on a circle. Circumscribed circle: a circle that contains the vertices of an inscribed polygon

  6. The measure of a central angle is equal to the measure of its intercepted arc. 1. For the circle with center G, name each of the following: a. three central angles: _______ , _______ , _______ A D G B C AGD DGC BGC

  7. two semicircles: _______ , _______ • three minor arcs: _______ , _______ , _______ • d. three major arcs: _______ , _______ , _______ A D G ADC ABC B BC DC C AB ABD DBC ADB

  8. A • 2. In the circle with center E, find • the measure of each angle or arc: • a. mBEC = _______o • m1 = _______o • m AC = _______o • d. m2 = _______o • e. m ADB = _______o 90° 1 2 80 D 50° E 50 90 B C 80° 140 140 360-220 220

  9. In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. If 1  2, then JK  KM . M If JK  KM, then 1  2. 1 2 K J

  10. A • STICKY NOTE PROBLEM: • In the circle with center E, find • the measure of each angle or arc: • a. mBEC = _______o • m1 = _______o • m AC = _______o • d. m2 = _______o • e. m ADB = _______o Sticky Note Problem 95° 1 2 D 30° E 70 30 B C 95 70° 125 165 360-195 195

  11. The measure of an inscribed angle is one half the measure of its intercepted arc. 3. In the diagram at right R and S are inscribed angles. mS = ½ m_______ = ½ ( ______o) = _______ o. m QS= 2m_______ = 2 ( ______o) = _______ o. R Q 37° 50° T S 25 RT 50 QRS 74 37

  12. Find the measure of each angle or arc: • a. mA = ____ b. mA = _____ • c. m BC = _____ d. m BC = _____ 70° 16° A 140° ½(32°) C ½(140°) B C B 32° A 180° 128° B A 2(64°) 2(90°) 64° A B C C

  13. Sticky Note Problem! • Find the measure of each angle or arc: • mA = ______ • m BC = _____ 62.5° 125° C ½(125°) B A 108° A 2(54°) 2(90°) Sticky Note Problem 54° B C

  14. An angle inscribed in a semicircle is a right angle. (Look at example #4d.) A quadrilateral can be inscribed in a circle, if and only if opposite angles are supplementary.

  15. Find the value of each variable: • a. x = _______, b. x = _______, • y = _______ y = _______ • c. x = _______, • y = _______ 92 15 80 30 88+ x = 180 x = 92 100+ y = 180 y = 80 3y° 88° 8x + 4x = 180 12x = 180 x = 15 3y + 3y = 180 6y = 180 y = 30 4x° 8x° 100° y° 3y° x° 18.5 18 4x + 106 = 180 4x = 74 x = 18.5 5y + 90 = 180 5y = 90 y = 18 4x° 5y° 106°

  16. Find the value of each variable: • a. x = _______, • y = _______ 165 105 15 + x = 180 x = 165 75 + y = 180 y = 105 15° 75° y° x° Sticky Note Problem

  17. 6. Determine whether a circle can be circumscribed about the quadrilateral. a. _________ b. _____________ c. _____________ NO YES 92° 90°+ 90° = 180° The sum of all angles of a quadrilateral = 360°. 360° – 180° = 180° 180° = 180° 130°+ 70 ° = 200° 200° ≠ 180° 130° 70° YES In isosceles triangles, base angles are congruent Parallel lines cut by a transversal – consecutive interior angles are supplementary 60° 60° 120° 120° a + 120° = 180° a = 60 60° + 120° = 180°

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