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Other Angle Relationships in Circles

Other Angle Relationships in Circles. Objectives/Assignment. Use angles formed by tangents and chords to solve problems in geometry. Use angles formed by lines that intersect a circle to solve problems. Using Tangents and Chords.

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Other Angle Relationships in Circles

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  1. Other Angle Relationships in Circles

  2. Objectives/Assignment • Use angles formed by tangents and chords to solve problems in geometry. • Use angles formed by lines that intersect a circle to solve problems.

  3. Using Tangents and Chords • 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. mADB = ½m

  4. Tangent to a Chord Conjecture • 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. m1= ½m 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°

  6. Ex. 2: 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°

  7. Ex. 3: Finding an Angle Measure (9x + 20)° • 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° 5x° D

  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 ) Chords intersecting Inside the circle • 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.

  13. m1 = ½ m( - m ) Tangent and Secant Exterior Intersections • 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.

  14. Tangent and Secant Exterior Intersections • 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 )

  15. Tangent and Secant Exterior Intersections • 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 )

  16. Ex. 4: 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

  17. mGHF = ½ m( - m ) Ex. 5: Tangent & Secant Intersections 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. 6: Tangent & Secant Intersections 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

  19. Ex. 7: 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. 7: 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. 7: 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°.

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