A ceiling fan has five equally spaced blades. What is the Repeat Exercise 27, but with the ceiling fan having six

T Q b A B C E F D t C O s a Figure 6.21 The term tangent also applies to a segment or ray that is part of a tangent line to a circle. In each case, the tangent touches the circle at one point. C O A B Q S V R T Figure 6.22 A secant is a line or segment or ray that intersects a circle at exactly two points. DEFINITION In Figure 6.21a, line s is a secant to ; also, line t is a tangent to and point C is its point of contact. In Figure 6.21b, is a tangent to and point T is its point of tangency; is a secant with points of intersection at E and F. CD }Q AB }O }O A polygon is inscribed in a circle if its vertices are points on the circle and its sides are chords of the circle. Equivalently, the circle is said to be circumscribed about the polygon. The polygon inscribed in a circle is further described as a cyclic polygon. DEFINITION In Figure 6.22, is inscribed in and quadrilateral RSTV is inscribed in . Conversely, is circumscribed about and is circumscribed about quadrilateral RSTV. Note that , , and are chords of and that , , and are chords of . Quadrilateral RSTV and are cyclic polygons. 䉭ABC }Q RV TV , ST RS } O AC BC AB }Q 䉭ABC } O }Q }O 䉭ABC Discover Draw any circle and call it . Now choose four points on in order, call these points A, B, C, and D. Join these points to form quadrilateral ABCD inscribed in . Measure each of the inscribed angles , , , and . a. Find the sum b. How are A and C related? c. Find the sum d. How are B and D related? ANSWERS ∠s m ∠B + m∠D. ∠s m ∠A + m∠C. ∠D ∠C ∠B ∠A }O }O }O a 180 b Supplementary c 180 d Supplementary If a quadrilateral is inscribed in a circle, the opposite angles are supplementary. Alternative Form: The opposite angles of a cyclic quadrilateral are supplementary. THEOREM 6.2.1 The preceding Discover activity prepares the way for the following theorem. The proof of Theorem 6.2.1 follows. In the proof, we show that and are supplementary. In a similar proof, we could also have shown that and are sup- plementary as well. ∠V ∠S ∠T ∠R