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12.4 Angle Measures in Circles

12.4 Angle Measures in Circles. Theorem 12.9. m ADB = ½m. Theorem 12.10. m 1= ½ m. m 2= ½ m. Ex. 1: Finding Angle and Arc Measures. Line m is tangent to the circle. Find the measure of the red angle or arc. Solution: m 1= ½ m 1= ½ (150 °) m 1= 75 °. 150°.

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12.4 Angle Measures in Circles

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  1. 12.4 Angle Measures in Circles

  2. Theorem 12.9 mADB = ½m

  3. Theorem 12.10 m1= ½m m2= ½m

  4. Ex. 1: Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m1= ½ m1= ½ (150°) m1= 75° 150°

  5. Ex. 1: Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the red angle or arc. • Solution: m = 2(130°) m = 260° 130°

  6. Ex. 2: Finding an Angle Measure • In the diagram below, is tangent to the circle. Find mCBD • Solution: mCBD = ½ m 5x = ½(9x + 20) 10x = 9x +20 x = 20  mCBD = 5(20°) = 100° (9x + 20)° 5x° D

  7. Lines Intersecting Inside or Outside a Circle • If two lines intersect a circle, there are three (3) places where the lines can intersect. on the circle

  8. Inside the circle

  9. Outside the circle

  10. Lines Intersecting • You know how to find angle and arc measures when lines intersect ON THE CIRCLE. • You can use the following theorems to find the measures when the lines intersect INSIDE or OUTSIDE the circle.

  11. m1 = ½ m + m m2 = ½ m + m Theorem 12.11 • If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

  12. m1 = ½ m( - m ) Theorem 12.11 • If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

  13. Theorem 12.11 • If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. m2 = ½ m( - m )

  14. Theorem 12.11 • If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 3 m3 = ½ m( - m )

  15. Ex. 3: Finding the Measure of an Angle Formed by Two Chords 106° • Find the value of x • Solution: x° = ½ (m +m x° = ½ (106° + 174°) x = 140 x° 174° Apply Theorem 10.13 Substitute values Simplify

  16. mGHF = ½ m( - m ) Ex. 4: Using Theorem 10.14 200° • Find the value of x Solution: 72° = ½ (200° - x°) 144 = 200 - x° - 56 = -x 56 = x x° 72° Apply Theorem 10.14 Substitute values. Multiply each side by 2. Subtract 200 from both sides. Divide by -1 to eliminate negatives.

  17. mGHF = ½ m( - m ) Ex. 4: Using Theorem 10.14 Because and make a whole circle, m =360°-92°=268° x° 92° • Find the value of x Solution: = ½ (268 - 92) = ½ (176) = 88 Apply Theorem 10.14 Substitute values. Subtract Multiply

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