Inscribes Angles and Polygons

Review: Remember the relationship between central angles and their arcs: Central angle = arc An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc.   Theorem: If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.   Inscribed angle = 1/2 of arc Example 1: Find the measures below. a. Find m∠NMP: 50° b. Find m∠WXY: 105° c. Find m∠DEF: 80° TRY: a. Find m∠BCD: 30° b. Find m∠KNP: 120° c. Find m∠KL: 20° Example 2: Find the measures below. Use everything you know about angles, triangles, and arcs! Given a circle with points A, B, C, D, and E, and various angle measures: a. Find m∠AB. m∠AB = 64° b. Find m∠C. m∠C = 180° - 85° - 32° = 63° c. Find m∠AXD. m∠AXD = 180° d. Find m∠B. m∠B = 85° + 32° = 117° 360° - 117° = 243° 243° / 2 = 121.5° m∠B = 121.5° e. Find mCD. mCD = 100° + 126° = 226° f. Find mBC. mBC = 360° - 290° = 70° Inscribed and Circumscribed If all the vertices of a polygon lie on a circle, then the polygon is inscribed in the circle and the circle is circumscribed about the polygon. Examples are given of inscribed and not inscribed polygons. If you inscribe a right triangle in a circle, then what is the measure of the arc intercepted by the right angle? 180° What must be true about the hypotenuse of the triangle? Hypotenuse is a diameter Theorem: If a triangle inscribed in a circle is a right triangle, then the hypotenuse is a diameter of the circle. If a side of a triangle inscribed in a circle is a diameter, then the triangle is a right triangle. Example 2: Find the values of x and y. a) x = 90° y = 40° b) x = 90° y = 30° Theorem: If a quadrilateral can be inscribed in a circle, then its opposite angles are supplementary. If the opposite angles of a quadrilateral are supplementary, then1 the quadrilateral can be inscribed in a circle.   Example 3: Find the values of the variables. a) z + 80° = 180° z = 100° y + 120° = 180° y = 60° b) g + 75° = 180° g = 105° h + 100° = 180° h = 80° TRY: Find the values of the variables. a) x = 90° y = 90° b) x = 40° y = 40° c) x = 85° d) x = 130°