# A more direct way to see that the angle inscribed in a semicircle is $90^\circ$?

Like this inscribed angle proof, another proof enabled by this clever angle sum proof.

Is there a simpler way to show this?

Is this proof original?

• What is your question here? – B. Mehta Apr 15 '18 at 3:16
• Sorry, "is it original?" (question edited) – hyperpallium Apr 15 '18 at 3:24
• Very unlikely to be original. But I do like it. – fleablood Apr 15 '18 at 4:18

## 1 Answer

Although this requires extending a line segment, I think it's easier to see that a $180^\circ$ line is divided in two (without algebraic rearrangment) - and the inscribed $90^\circ$ angle is right there, half of the line.

Also, the corresponding angle used is arguably simpler than the alternate interior angle in the question.

Regarding presentation, the lighter coloured angles are more legible. I think moving the inscribing angle nearer to the center would be better, to make the red and blue angles closer in measure, and therefore more equally visible. But I'd have to redo most of it, and it's a time-consuming nightmare making angles colours go between the segments you want in geogebra/classic (it sometimes prefers the other end of the segment, as if extended).

• This is good--the 180 divided into two equal blues plus two equal reds, so that the angle made by one of each must be right. But to see that the two reds are in fact equal don't you have to color in an alternate interior red at the center, which we can recall is equal to the sum of the two reds in the isosceles triangle (Euclid I,32)? – Edward Porcella Apr 24 '18 at 15:36
• Thanks! The upper of the pair of reds is an alternate interior angle to the red at the bottom-right. (Different colours for the isosceles angles would help here too). Oh! I commented that I used I, 32 here, but it's not true (though it can be seen that way). I did use it in way: the lopsidedness of I, 32 made me think of extending a line and using a corresponding angle, as it does. (I'd wanted to use that top-most blue angle before, but perhaps strangely, I couldn't see that it was simply a corresponding angle to the bottom-left blue... I, 32 loosened me up enough to see it). – hyperpallium Apr 25 '18 at 4:32
• @EdwardPorcella sorry, without the @, you wouldn't get notification. – hyperpallium Apr 26 '18 at 4:47