Proving Angles in a Circle: Angle 2 is Twice Angle 1

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SUMMARY

The discussion focuses on proving that angle 2 is twice the size of angle 1 in a circle. The method involves drawing a line segment from the center of the circle to a peripheral point, which divides angle 1 into two angles, alpha and beta. By analyzing the isosceles triangles formed and applying the angle sum properties, participants derive the relationship between angle 1 and angle 2. The conclusion is that if the two chords from the peripheral point are equal, then the equation 2(pi - angle 1) + angle 2 = 2 pi holds true, confirming the relationship.

PREREQUISITES
  • Understanding of basic circle geometry
  • Knowledge of isosceles triangles
  • Familiarity with angle sum properties
  • Ability to manipulate algebraic expressions involving angles
NEXT STEPS
  • Study the properties of isosceles triangles in circle geometry
  • Learn about angle subtended at the center of a circle
  • Explore the concept of angle sums around a point
  • Investigate the relationships between angles formed by intersecting chords
USEFUL FOR

Students studying geometry, mathematics educators, and anyone interested in understanding the properties of angles in circles.

Vashti
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http://s14.photobucket.com/albums/a348/Drakhys-2/?action=view&current=scan.jpg

How would I prove that angle 2 is twice the size of angle 1?


thankyou,
Vash.
 
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Hint : join the center to the peripheral point with a line segment. Let that line segment divide angle 1 into two (not necessarily equal) angles alpha and beta. Now observe the two isoceles triangles formed by the dividing line segment. By considering angle sums, find the angle in each triangle that is subtended at the center of the circle. Now using the angle sum about a point, find angle 2 in terms of alpha and beta, and hence angle 1.
 
If the two chords from the top point are of equal length (?) then can you see that 2(pi -angle 1) + angle 2 must equal 2 pi?
 

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