Understanding Chords and Inscribed Angles in Circles

Explore the relationships between central angles, chords, arcs, and inscribed angles in circles through theorems and examples. Learn about the Inscribe Angle Theorem and its corollaries to deepen your understanding of circle geometry concepts.

• Circles
• Geometry
• Inscribed Angles
• Chords
• Theorems

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1. NO WARMUP Copy thm 12.9, corollaries, and thm 12.10 from pgs 679 and 680

2. 12.2/12.3: CHORDS AND ARCS & INSCRIBED ANGLES LEQ: WHAT ARE THE THEOREMS INVOLVED AND CALCULATIONS WITH CHORDS AND ARCS?

3. THEOREM SHEET Thm. 12.4: 1.) Congruent central angles have congruent chords (vice versa) 2.) Congruent chords have congruent arcs 3.) Congruent arcs have congruent central angles

4. NOTES: PROOF OF THM. 12.4 In the circle on the right, prove if m<CAD=m<FAE, the CD=FR. C D F A E

5. THEOREM SHEET Ex. 3: Find AB.

6. All go hand in hand: Perp, bisect, and diameter: one makes all the others true.

7. P and Q are points on O. The distance from O to PQ is . Ex. 4 15 in., and PQ = 16 in. Find the radius of O. . More examples: Find the missing lengths.

8. VOCAB: INSCRIBED ANGLE -an angle whose vertex is on a circle and whose sides are chords <ABC and <DEF are inscribed angles in the circles shown below: <ABC intercepts minor arc AC <DEF intercepts major arc DGF *intercept means forms

9. THEOREM 12-9: INSCRIBED ANGLE THEOREM A The measure of an inscribed angle is equal to half of its intercepted arcs. C B 1 = m B m A C 2

10. EXAMPLES Find m<ABC Find m<ABC and m<ABD

11. EXAMPLES Find m<DEF and mAEC Find m<ABC

12. 3 INSCRIBED < COROLLARIES If two inscribed angles intercept the same arc, then the angles are congruent. An angle inscribed in a semicircle is a right angle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

13. THEOREM 12-10 The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. 1 = m C mBDC 2 B B D D C C

14. EX: FIND THE NUMBERED ANGLES. 108 1 32 1 2 1 2 102 1 61 46

15. HWK: Copy thms 12.11 and 12.12