Answered: "One of the rays of the inscribed angle..."

Licinilg

Open question

2022-08-24

One of the rays of the inscribed angle is the diameter.
Given: Circle O
BC is a diameter.
Prove:

m ∠ A C B = \frac{1}{2} m A B

Kristen Garrison

Beginner2022-08-25Added 11 answers

Step 1
The case you drew is perhaps the easiest one:

Δ A O C

is isosceles, with

A O = O C ⟹ ∠ A C O = ∠ C A O

, and since

∠ A O B

is an external angle to triangle

Δ A O C

.
Step 2
Then it equals the sum of the two triangle's
angles whose vertex it doesn't share, thus

∠ A O B = ∠ C A O + ∠ A C O = 2 ∠ A C O

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