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Understanding Angles and Arcs in Circles: Theorems and Applications

This lesson covers essential theorems regarding angles and arcs in circles, focusing on the relationships involving tangents, chords, and secants. Students will learn that the measure of an inscribed angle is half that of its intercepted arc. We will explore various cases, including when lines intersect inside or outside a circle, using Theorem 10.12 and Theorem 10.14 to find unknown angle measures. Through practical examples and problems, students will apply these concepts to determine arc and angle measures and reinforce their understanding through classroom activities.

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Understanding Angles and Arcs in Circles: Theorems and Applications

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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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