10-3 Arcs and Chords A chord is a segment with ___________________________________. If a chord is NOT a diameter, then its endpoints divide the circle into a major arc and a minor arc. Theorem 10.2 In the same circle or in congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent. Example: FG ≅ HJ IFF FG ≅ HJ L F J G H

Real-World Example 1: Use congruent chords to find arc measure In the embroidery hoop, AB ≅ CD and m AB = 60. Find m CD. 60o A C B D

Example 2: Use congruent arcs to find chord lengths In the figures, circle J is congruent to circle K and MN ≅ PQ. Find PQ N M 2x+1 3x-7 P Q J K

Bisecting Arcs and Chords If a line, segment, or ray divides an arc into two congruent arcs, then it bisects the arc. Theorems: 10.3 If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. Example: If diameter AB is perpendicular to chord XY, then XZ ≅ ZY and _______________ The perpendicular bisector of a chord is a diameter(or radius) of the circle. Example: If AB is a perpendicular bisector of chord XY, then AB is a diameter of circle C. B Y X C A Z B Y X C A

Example 3: Use a radius perpendicular to a chord In circle S, m PQR = 98. Find m PQ R S Q P T 6

Real-World Example 4: Use a diameter perpendicular to a chord. In a stained glass window, diameter GH is 30 inches long and chord KM is 22 inches long. Find JL. M H G L K J

Theorem 10.5 In the same circle or in congruent circles, two chords are congruent IFF they are equidistant from the center. Example: FG ≅ JH IFF LX = LY G H J F L X Y Example 5 : Use chords equidistant from center. In circle A, WX = XY = 22. FInd AB. Y C X W B 5x 3x+4 A

10-4 Inscribed Angles An inscribed angle has a vertex on a circle and sides that contain chords of the circle. T he Inscribed Angle Theorem says that an inscribed angle is half the measure of it's intercepted arc. Example: Circle C has an inscribed angle of : C S Q R An intercepted arc has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle. In circle C, minor arc QS is intercepted by <QRS. Case 1Case 2Case 3 Center P is on a side of the inscribed angle. P P P Center P is inside the inscribed angle Center P is in the exterior of the inscribed angle

Example 1: Use inscribed angles to find measures Find each measure. a. m<P b. m PO 56o 70o M O P N

A D F C 98o 40o Find m CF m <C

Theorem 10.7 If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. Example: <B and <C both intercept AD. SO, <B ≅ <C C D A B Example 2: Use inscribed angles to find measures Find m<T. T U V S (2x+15)o (3x-5)o

Angles of Inscribed Polygons Theorem 10.8 An inscribed angle of a triangle intercepts a diameter or a semicircle IFF the angle is a right angle. Example: If FJH is a semicircle, then m <G = 90. If m<G = 90, then FJH is a semicircle and FH is a diameter. H J G F Example 4: Find angle measures in inscribed triangles Find m<F G H F J (4x+2)o (9x-3)o ΔFGH is a right triangle because < G _____________________________. inscribes a semicircle

Theorem 10.9 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Example: If quadrilateral KLMN is inscribed in circle A, then < L and < N are supplementary and < K and < M are supplementary. M N L K A Real-World Example 5: Find angle measures The necklace charm shown uses a quadrilateral inscribed in a circle. Find m<A and m<B. B A D C xo (2x-30)o