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10.4 Other Angle Relationships in Circles

10.4 Other Angle Relationships in Circles. Learning Target. I can use theorems about tangents, chords and secants to solve unknown measure of arcs and angles. Review on inscribe angles.

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10.4 Other Angle Relationships in Circles

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  1. 10.4 Other Angle Relationships in Circles

  2. Learning Target • I can use theorems about tangents, chords and secants to solve unknown measure of arcs and angles .

  3. Review on inscribe angles • You know that measure of an angle inscribed in a circle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle. n x mADB = ½m Angle x = ½ n

  4. Theorem 10.12 • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. n x m1= ½m Angle x = ½ n m2= ½m

  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1= ½ m1= ½ (150°) m1= 75° 150° Angle x = ½ n

  6. 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° Angle x = ½ n

  7. 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 Angle x = ½ n

  8. 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

  9. Inside the circle

  10. Outside the circle

  11. 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.

  12. m1 = ½ m + m m2 = ½ m + m Theorem 10.13 n x • 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. f Angle x = ½ ( f + n)

  13. m1 = ½ m( - m ) Theorem 10.14 • 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. x n f Angle x = ½ ( f - n )

  14. Theorem 10.14 • 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. x n f Angle x = ½ ( f - n ) m2 = ½ m( - m )

  15. Theorem 10.14 • 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. x 3 n f Angle x = ½ ( f - n ) m3 = ½ m( - m )

  16. 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° Angle x = ½ (f +n) 174° Apply Theorem 10.13 Substitute values Simplify

  17. mGHF = ½ m( - m ) Ex. 4: Using Theorem 10.14 Angle x = ½ (f – n) 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.

  18. 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 Angle x = ½ ( f – n )

  19. Ex. 5: Describing the View from Mount Rainier • You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level. Find the measure of the arc that represents the part of Earth you can see.

  20. Ex. 5: Describing the View from Mount Rainier • You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level. Find the measure of the arc that represents the part of Earth you can see.

  21. Ex. 5: Describing the View from Mount Rainier • and are tangent to the Earth. You can solve right ∆BCA to see that mCBA  87.9°. So, mCBD  175.8°. Let m = x° using Trig Ratios

  22. 175.8  ½[(360 – x) – x] 175.8  ½(360 – 2x) 175.8  180 – x x  4.2 Apply Theorem 10.14. Simplify. Distributive Property. Solve for x. From the peak, you can see an arc about 4°.

  23. Reminders: • Pair-Share: Work on page. 624-625 #2-35 • Refer to the summary sheet “Angles Related to Circles” to identify what formula to use. • Quiz on Friday about Trigonometry and Circles.

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