Clock Activity 10.4

1. Clock Activity 10.4 • Please begin this activity • In this activity, you will use the degrees of a clock to make an conjecture about inscribed angles.

2. Clock Activity 10.4 2 *Angle measure Created = Arc Measure 2 *Angle measure Created = Arc Measure Inscribed angles whose endpoints are the diameter form right angles

3. 10.4Use Inscribed Angles and Polygons

4. Theorems 10-7: If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. What do you think? - If a second angle intercepted the same arc? 10-8: If two inscribed angles of a circle or congruent circles intercept congruent arcs of the same arc, then the angles are congruent.

5. Theorems What do you think? - If an angle intercepted a semicircle? 10-9: If an inscribed angle of a circle intercepts a semicircle, then the angleis a right angle.

6. Theorems 10-10: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. What do you think? - With an inscribed angle, what do you think the opposite angles would equal? Why?

7. R S 1 Q T 2 P Class Exercises • 1. Explain how an intercepted arc and an inscribed angle are related. • 2. ∆ABC is inscribed in a circle so that BC is a diameter. What type of triangle is ∆ ABC? Explain your answer. • 3. In the circle at the right, mST = 68. Find the m<1 and m<2 . The inscribed angle measures half of the intercepted arc. Since the inscribed angle intercepts in a semicircle, the angle is a right angle. Therefore, the triangle is a right triangle. m<1 = 34 m<2 = 34

8. Q R A 3 4 P S 1 2 T Class Exercises • In circle A: • and PQ  RS. Find • the measures of • angles 1,2,3 and 4. 3y – 9 = 4y - 25 6x + 11 + 9x + 19 = 90 y = 16 15x + 30 = 90 x = 4 m<1 = 35, m<2 = 55, m<3 = 39, and m<4 = 39

9. Clock Activity 10.5 • In this activity, you will use the degrees of a clock to make an conjecture about different types of angle relationships in circles using tangents, chords, and secants.

12. 10.5Apply Other Angle Relationships in Circles

13. Terms: Secant: A line that intersects a circle in exactly two places. It contains a chord of the circle. Theorems: If a _____________ and a _______________ intersect at the point oftangency, then the measure of each angle formed is _______________ the measure of its intercepted ______________. If m<RTW = 50, then arc RT = _________ If arc RT = 86, then m<RTW = _________ and m<RTV = ________. secant tangent one-half arc 100 43 137 Activity I Intercepted Arcs

14. interior secants If two _________________ intersect in the ______________ of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the ___________ and its ______________ angle. Similar to finding the average of the arc measures. Formula: ______________________ If arc ST = 30 and arc RU = 80 then If arc ST = 30 and m<1 = 50, m<3 = ______ and m<2 = _______ then arc RU = ______ angle vertical 55 125 70 Activity II Chords and Angles

15. Activity III Secants and angles If two secants, a secant and a tangent, or two tangents intersect in the ___________ of a circle, then the measure of the angle formed is __________ the positive difference of the measures of the ___________________ arcs.Case 1: two secants Case 2: one secant / Case 3: 2 tangents one tangent one-half exterior intercepted

16. Examples 1. In Circle Q, m<CQD = 120, mBC = 30, and m<BEC = 25. Find each measure. a. mDC b. mAD c. mAB d. m<QDC 30 120 80 130 2. Use Circle K to find the value of x. <R is formed by a secant and a tangent. x = 25

17. Examples 3. Use Circle S to find the value of y. y = 152 126.5° x° 85° 22°

18. Classwork • Finish ws

19. THAT'S ALL FOR TODAY!