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12 .5 Inscribed Angles and Triangles Recall: A Central Angle has the same measure as the arc in intercepts. .100° An inscribed 100° angle is not equal to the arc it intercepts. B Chords Inscribed Angle - An angle in a circle whose vertex is on the circle and sides contain cords of the circle. A C Intercepted Arc •Ex: AC •Measured in degrees • Ex: ABC . P 23° 46° What do you notice about the measures of the central and the inscribed angles? An inscribed angle is always half of the central angle with the same intersected arc. measure of = ½ (measure of central angle) inscribed angle x° (½)x° I think of an inscribed angle as being farther away from the arc, so it is smaller. Ex 1 - Find mDOG, mDUG, and mDIG given that the measure of arc DG is 50°: mDOG = 50° 50° G since it is a central D angle mDUG = ½ of 50° = 25° since it is an O inscribed angle mDIG = 25°, since it is also and inscribed angle I U Ex.1: Find the measure of angle b and arc a given that mPQT = 60° and m TS = 30°. mQ = ½m PT P a° = 120° 60 = ½m PT 120 = m PT T 60° 30° Q mb = ½m PS mb = ½(120 + 30) mb = 75° S b° R Ex 2 – Find the mDOT The arc intercepted by DFT is double mDFT D So, mDOT = 24° O ? F 24° 48° T Corollary 1 for Inscribed Angle • Two inscribed angles that intercept the same arc are congruent. B 100° 50° A 50° D C Ex 3 - Find the mTAP, given that OT = 80° and OP = 110°. If you add TO and OP, you get 190°. O 80° 110° T P A Since TAP is an inscribed angle, divide the arc in half to get: mTAP = 95° Corollary 2 for Inscribed • An inscribed in a semicircle is a right . 180° 90° Ex 4 – Find mRAT given that RA measures 70°. TA is a 70° A 55° . R 35° T Angles in a triangle add up to 180° . Q diameter since it passes through the center. TQA measures 180°, so mTRA = 90° Ex 5 – find x: Because a side passes through the center, the angle opposite the hypotenuse is 90°. x²+(3x)² = 10² 10 1x²+ 9x² = 100 . 10x² = 100 3x x x² = 10 x = √10 Example: • 2(68) = 136 = XW • 80 + 136 + ? = 360 • ? = 144 Example: • • • • • • • • • • 2(32) = 64 = YZ YZ + XY = 180 64 + 15x + 11 = 180 75 + 15x = 180 15x = 105 x=7 Or 90 + 32 = 122 180 – 122 = 58 15x + 11 = 116 x=7 Example: • 2(4x–5) = CDN • 8x – 10 + 6x + 10 + 150 =360 • 14x = 360 – 150 • 14x = 210 • x = 15 Example: • • • • • • • 3x + 23 + 9x + 1 = 180 12x + 24 = 180 12x = 156 x = 13 3(13) + 23 = 62 TS = 62 Angle TRS = 31 Example: • • • • • • • • • YRS = 2 (2 + 49x) 4 + 98x + 49x + 2 + 31x – 2 4 + 178x = 360 178x = 356 x=2 2 + 49(2) = 100 Check: 100 + 200 + 60 = 360 YRS = 200 TY = 49(2)+2 =100 TS = 31(2) – 2 = 60 What have we learned?? • The measure of a central angle = the measure of its intercepted arc. • The measure of an inscribed angle = ½ the measure of its intercepted arc. 50° A 60° 120° 120° 100° If the hypotenuse of an inscribed triangle is the diameter of the circle, then it is a right triangle. . Then you can use the fact that angles in a triangle add up to 90°, and inscribed angles are half of their intercepted arcs to find missing angles and arcs.