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Question Number 184655 by Mastermind last updated on 10/Jan/23

prove that an angle inscribe in a semi−circle is a right angle. Help!

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{inscribe}\:\mathrm{in}\:\mathrm{a}\: \\ $$$$\mathrm{semi}−\mathrm{circle}\:\mathrm{is}\:\mathrm{a}\:\mathrm{right}\:\mathrm{angle}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Answered by HeferH last updated on 10/Jan/23

Commented by HeferH last updated on 10/Jan/23

AB^(⌢) = 180° ∠ACB = ((AB^(⌢) )/2) = 90° (inscribed angle theorem)

$$\overset{\frown} {{AB}}\:=\:\mathrm{180}° \\ $$$$\:\angle{ACB}\:=\:\frac{\overset{\frown} {{AB}}}{\mathrm{2}}\:=\:\mathrm{90}°\:\left({inscribed}\:{angle}\:{theorem}\right) \\ $$

Commented by HeferH last updated on 10/Jan/23

Commented by HeferH last updated on 10/Jan/23

2α + 2θ + (180°−2θ − β) = 180° β = 2α

$$\mathrm{2}\alpha\:+\:\mathrm{2}\theta\:+\:\left(\mathrm{180}°−\mathrm{2}\theta\:−\:\beta\right)\:=\:\mathrm{180}° \\ $$$$\:\beta\:=\:\mathrm{2}\alpha\: \\ $$

Commented by Mastermind last updated on 10/Jan/23

What is this one again?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{this}\:\mathrm{one}\:\mathrm{again}? \\ $$

Commented by Mastermind last updated on 10/Jan/23

Thank you BOSS

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{BOSS} \\ $$

Commented by HeferH last updated on 10/Jan/23

Proof of theInscribed angle theorem :)

$$\left.{Proof}\:{of}\:{theInscribed}\:{angle}\:{theorem}\::\right) \\ $$

Commented by Mastermind last updated on 10/Jan/23

Thank you BOSS but i think the first one is the corrct solution

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{BOSS}\:\mathrm{but}\:\mathrm{i}\:\mathrm{think}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{one}\:\mathrm{is}\:\mathrm{the}\:\mathrm{corrct}\:\mathrm{solution} \\ $$

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