Warm-Up: 1)simplify: (x – 1)² 2)Factor: 10r² – 35r 3) Factor: t ² + 12t ) Solve: 2x ² – 3x + 1 = x ² + 2x – 3 5) Find the radius of a circle with a 30 cm chord 20 cm away from the center of the circle.
11.3: Inscribed Angles
Inscribed Angle Angle with its vertex on a circle and sides are chords of the circle DEF is inscribed in G DEF intercepts DF DF is intercepted by DEF
Inscribed Angle Theorem: The measure of an inscribed angle is one-half the measure of its intercepted arc. (or arc measure is twice inscribed angle measure) If DF=76°, then m DEF=38°
Corollary 1: Two inscribed angles that intercept the same arc are congruent. 1 & 6 intercept WX, so 1 6 If m 1=47°, then WX=94°, and m 6=47°
Corollary 2: An angle inscribed in a semicircle is a right angle. BD is a diameter BCD is a semicircle, so BAD = 90° BAD is a semicircle, so BCD = 90°
Corollary 3: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Quadrilateral EFGH is inscribed in N E + G = 180 F + H = 180 If m E=125°, If m F=87°, then m G=55° then m H=93°
Theorem 11-10: The measure of an angle formed by a tangent and a chord is one-half the measure of the intercepted arc. W Z X Y V 115° 245°
11.4: Angle Measures
Secant A line that intersects a circle in exactly 2 points Line AB is a secant to N A B N
Theorem 11.11a: The measure of an angle formed by 2 lines intersecting inside of a circle is ½ the sum of the measures of its intercepted arcs. A B C D 164°22°
Theorem 11.11b: The measure of the angle formed by 2 lines intersecting outside of a circle is ½ the difference of the measures of the intercepted arcs. i) 2 secants: D B C A E 72° 15° x°
Theorem 11.11b: The measure of the angle formed by 2 lines intersecting outside of a circle is ½ the difference of the measures of the intercepted arcs. ii) secant-tangent: K L J M 146° 54° x°
Theorem 11.11b: The measure of the angle formed by 2 lines intersecting outside of a circle is ½ the difference of the measures of the intercepted arcs. iii) 2 tangents: S R T U 266° x°