Proving the Secant Angles in the CircleHow to prove the theorem of angles in the circle. Like this image here:How prove that the angle X is the half of the sum of both angles' measurement of the Arc AC and Arc BD? This image here:How prove that the angles A is the half of the difference of both angle measurements of Arc BC and Arc DE?

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Proving the Secant Angles in the CircleHow to prove the theorem of angles in the circle. Like this image here:How prove that the angle X is the half of the sum of both angles&#039; measurement of the Arc AC and Arc BD? This image here:How prove that the angles A is the half of the difference of both angle measurements of Arc BC and Arc DE?

Paulina Horne · Accepted Answer

Step 1In the first case look at the triangle     Δ  B  C  K    ,      K   the point of intersection of both secants. Observe that   ∠  x  =      180    ∘    −  ∠  D  C  B  −  ∠  A  B  C.Step 2But   ∠  D  C  B  =      1    2    ∠  B  M  C  =      1    2                      B        C            ^          ,      ∠  A  B  C  =      1    2    ∠  A  M  C  =                    A        C            ^      .

opositor5t · Accepted Answer

Step 1For the second case.Draw             B      E        ¯   and let the measures of arcs CB and ED be   α and   β.By the inscribed angle theorem, the measure of an angle inscribed in a circle is half the measure of its intercepted arc.Therefore   ∠      E    =      α    2   and   ∠      E    B    D    =      β    2  Step 2By the exterior angle theorem   m  ∠      A    +  m  ∠      E    =  m  ∠      E    B    D  Substituting   m  ∠      E    =      β    2    −      α    2  .

How to prove the theorem of angles in the circle. How can I prove that the angle X is the half of the sum of both angles' measurement of the Arc AC and Arc BD? How can I prove that the angles A is the half of the difference of both angle measurements of Arc BC and Arc DE?

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