Given: in Given: in which

In Exercises 32 to 37, write a paragraph proof.

32. Given: and are

diameters of Prove:

33. Given: Chords , , ,

and in Prove: 䉭ABE 䉭CDE }O AD CD BC AB 䉭RST ⬵ 䉭VTS }W TV RS R V W T S A C O B D E M N O Q P S T Q V

34. Congruent chords are located at the same distance from

the center of a circle.

35. A radius perpendicular to a chord bisects the arc of that

chord. 36. An angle inscribed in a semicircle is a right angle. 37. If two inscribed angles intercept the same arc, then these angles are congruent.

38. If in

, explain why MNPQ is an isosceles trapezoid. HINT: Draw a diagonal. }O Í MN 7 Í PQ B C A O Y O X Z W

39. If , explain why

is an isosceles triangle. 䉭STV ST ¬ ⬵ TV ¬

40. Use a paragraph proof to complete this exercise.

Given: with chords and , radii and Prove: m ∠ABC 6 m∠AOC OC AO BC AB }O

41. Prove Case 2 of Theorem 6.1.2. 42. Prove Case 3 of Theorem 6.1.2.

43. In , and

. If find WZ . XW ⬵ WY, XZ = 6 OY = 5 }O Tangent Point of Tangency Secant Polygon Inscribed in a Circle Cyclic Polygon Circumscribed Circle Polygon Circumscribed about a Circle Inscribed Circle Interior and Exterior of a Circle More Angle Measures in the Circle 6.2 KEY CONCEPTS We begin this section by considering lines, rays, and segments that are related to the circle. We assume that the lines and circles are coplanar. A tangent is a line that intersects a circle at exactly one point; the point of intersection is the point of contact, or point of tangency. DEFINITION 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