Geometry - Inscribed Angles - same arc

Prove that:

If two inscribed angles intercept the same arc, then the angles are congruent.

Note the following definition:
• to be an inscribed angle in a circle, the vertex must be on the circle.
• the two sides of the angle are chords which extend from the vertex to another point on the circle

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Comments for Geometry - Inscribed Angles - same arc

 Jul 29, 2013 Proof by: Staff Answer Part I "all" inscribed angles in a circle are always equal to 1/2 the intercepted arc Therefore, if any two inscribed angles intercept the same arc, both angles are exactly 1/2 the same arc. Therefore, the two inscribed angles must equal one another. The can be proven by showing it is true for three general cases: Case I: If one of the chords of the inscribed angle passes through the center of the circle: ------------------------------------

 Jul 29, 2013 Proof by: Staff ------------------------------------ Part II ∠x + ∠x + 180° - ∠y = 180° ∠x + ∠x + 180° - 180° - ∠y = 180° - 180° ∠x + ∠x + 0 - ∠y = 0 ∠x + ∠x - ∠y = 0 ∠x + ∠x - ∠y + ∠y = 0 + ∠y ∠x + ∠x + 0 = 0 + ∠y ∠x + ∠x = ∠y 2∠x = ∠y 2∠x / 2 = ∠y / 2 ∠x = ½∠y If one of the chords of the inscribed angle passes through the center of the circle, the inscribed angle is always equal to ½ of the central angle that subtends the same arc. Case II: If the center of the circle falls between the two chords of the inscribed angle: ------------------------------------

 Jul 29, 2013 Proof by: Staff ------------------------------------ Part III using the results of case I, above ∠x₁ = ½∠y₁ ∠x₂ = ½∠y₂ ∠x₁ + ∠x₂ = ½∠y₁ + ½∠y₂ ∠x₁ + ∠x₂ = ½(∠y₁ + ∠y₂) since ∠x = ∠x₁ + ∠x₂ ∠y = ∠y₁ + ∠y₂ then ∠x = ½(∠y) If the center of the circle falls between the two chords of the inscribed angle, the inscribed angle is always equal to ½ of the central angle that subtends the same arc. Case III: If the center of the circle falls outside the two chords of the inscribed angle: using the results of case I and case II, above ∠x₂ = ½∠y₂ ∠x = ½∠y ∠x₁ + ∠x₂ = ½(∠y₁ + ∠y₂) since ∠x₂ = ½∠y₂ ∠x₁ + ½∠y₂ = ½(∠y₁ + ∠y₂) ∠x₁ + ½∠y₂ = ½∠y₁ + ½∠y₂ ∠x₁ + ½∠y₂ - ½∠y₂ = ½∠y₁ + ½∠y₂ - ½∠y₂ ∠x₁ + 0 = ½∠y₁ + 0 ∠x₁ = ½∠y₁ Conclusion: Regardless of where the center of the circle falls (on one of the chords of the inscribed angle, between the chords of the inscribed angle, or outside the chords of the inscribed angle), the inscribed angle is always equal to ½ of the central angle that subtends the same arc. Thanks for writing. Staff www.solving-math-problems.com