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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


THANK YOU!! I NEED HELP HAHA

Comments for Geometry - Inscribed Angles - same arc

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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:



Center of a circle





 Case I:  one of the chords of the inscribed angle passes through the center of the circle




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Jul 29, 2013
Proof
by: Staff


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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:




Case II:  the center of the circle falls between the two chords of the inscribed angle




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Jul 29, 2013
Proof
by: Staff


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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:



Case III:  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
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