Lesson 6.3 Inscribed Angles and their Intercepted Arcs

1. Lesson 6.3 Inscribed Angles and their Intercepted Arcs Objectives: Using Inscribed Angles Using Properties of Inscribed Angles. Homework: Lesson 6.3/ 1-12 Friday-Chapter 6 Quiz 2 on 6.1-6.3

2. Using Inscribed Angles Inscribed Angles & Intercepted Arcs An INSCRIBED ANGLE is an angle whose vertex is on the circle and whose sides are chords of a circle. ∠ABC is an inscribed angle

3. Using Inscribed Angles Measure of an Inscribed Angle

4. Using Inscribed Angles Example 1: Find the mand mPAQ . = 2 * m PBQ = 2 * 63 = 126˚ 63°

5. Using Inscribed Angles Example 2: Find the measure of each arc or angle. Q = ½ 120 = 60˚ = 180˚ R = ½(180 – 120) = ½ 60 = 30˚

6. Using Inscribed Angles Inscribed Angles Intercepting Arcs Conjecture If two inscribed angles intercept the same arc or arcs of equal measure then the inscribed angles have equal measure. mCAB = mCDB

7. Example 3: Using Inscribed Angles Find =360 – 140 = 220˚

8. Example 4: Find mCAB and m Using Properties of Inscribed Angles mCAB = ½ mCAB = 30˚ m = 2* 41˚ m = 82˚

9. Using Properties of Inscribed Angles Cyclic Quadrilateral A polygon whose vertices lie on the circle, i.e. a quadrilateral inscribed in a circle. Quadrilateral ABFE is inscribed in Circle O.

10. Using Properties of Inscribed Angles Cyclic Quadrilateral Conjecture If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. m∠A + m∠C = 180° m∠B + m∠D = 180°

11. Using Properties of Inscribed Angles Example 5: Opposite angles of an inscribed quadrilateral are supplementary Find the measure of Intercepted arc of an inscribed angles = 2* angle measure

12. Example 6: Find m∠A and m∠B Opposite angles of an inscribed quadrilateral are supplementary m∠A + 60° = 180° m∠A = 120° m∠B + 140° = 180° m∠B = 40°

13. Example 7: Using Properties of Inscribed Angles Find x and y Opposite angles of an inscribed quadrilateral are supplementary

14. A polygon is circumscribed about a circle if and only if each side of the polygon is tangent to the circle. Using Properties of Inscribed Angles Circumscribed Polygon

15. Using Properties of Inscribed Angles Angles inscribed in a Semi-circle Conjecture A triangle inscribed in a circle is a right triangle if and only if the diameter is the hypotenuse A has its vertex on the circle, and it intercepts half of the circle so that mA = 90.

16. Example 8: Angles inscribed in a semi-circle are right angles Find x.

17. Example 9: Using Inscribed Angles 146° FindmFDE

18. Using Properties of Inscribed Angles Parallel (Secant) Lines Intercepted Arcs Conjecture Parallel (secant) lines intercept congruent arcs. X A Y B

19. Example 10: Using Properties of Inscribed Angles Find x. 360 – 189 – 122 = 49˚ x 122˚ x = 49/2 = 24.5˚ x 189˚

20. Tangent/Chord Conjecture The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. B B D D C C

21. Example 11:

22. Example 12: Using Tangent/Chord Conjecture Triangle sum Find x and y. J 90o Q 35o yo 55o xo L K

23. Homework: Lesson 6.3/ 1-12