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?

ferdysy9

ferdysy9

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2022-08-17

Proving the Secant Angles in the Circle
How 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?

Answer & Explanation

Paulina Horne

Paulina Horne

Beginner2022-08-18Added 10 answers

Step 1
In 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 2
But D C B = 1 2 B M C = 1 2 B C ^ , A B C = 1 2 A M C = A C ^ .
opositor5t

opositor5t

Beginner2022-08-19Added 5 answers

Step 1
For 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 2
By the exterior angle theorem m A + m E = m E B D
Substituting m E = β 2 α 2 .

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