Can someone help me with these 2 Euclidean geometry questions?

Hi Ameer Bux:

Problem 7 is tricky because the figure is not drawn in a manner consistent with the statements about it. You might find it helpful to redraw it so that SB is a diameter of the circle ABC. At least that's how I interpret the text:"SB bisects ABC."

Problem 8 is easier. I suggest you look up

https://en.wikipedia.org/wiki/Exterior_angle_theorem .​

Read about proposition 1.32.

Hope this helps.

Regards,
Buzz

 

What do the numbers (1&2) in the diagram indicate? Are all angles labeled 1 supposed to be equal?

 

I think it might mean that line SB bisects angle OBC. Here's a picture drawn to scale assuming that:

 

@Ameer Bux I have verified that the figure should look like I drew in the post above with SB bisecting angle ABC. As a hint towards a proof I would remind you that angles on circles that subtend equal arcs are equal.

 

Ameer appears to have disappeared, which is annoying enough. But to answer your question, looking at his second picture, I think when he as a 1 and 2 at vertex A that it is a very awkward notation where it would have been much better to call them A1 and A2. Similarly the 1 and 2 at vertex R refer to angles better notated as R1 and R2. So all those 1's and 2's are different. Awful notation in his pictures.

 

Buzz, do you have an argument for problem 8? I don't see it and I have my doubts it is even true. And I don't see proposition 1.32 at that link. ??

 

Hi LCKurtz:

Thank you for noting my error. Somehow I read the handwritten question as referring to R₁ rather than B₁. Sorry for my carelessness. Just another of my senior moments.

From the link:

... the term "exterior angle theorem" has been applied to a different result,[1] namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.​

Regards,
Buzz

​

 

Hi Buzz. I wasn't trying to point out errors though, I'm really wondering if his Problem 8 is true. So I was hoping you had a proof. I can't figure out a proof and I drew what I consider to be a reasonably accurate figure which leads me to believe it may be false.

 

Maybe I am missing something but it looks quite easy.
I sent details to Buzz in a private conversation, but for some reason it would not let me add you as a recipient. You must be in a higher astral plane. I thought it premature to post what might be a solution to the thread.

@Ameer Bux , what do you know about opposite angles of a quadrilateral whose vertices lie on a circle?

 

For the first problem, the theorem which states that equal arcs subtend equal angles on a circle is applicable.

 

I really have to learn geometry. I look foolish with a math degree without knowing geometry

 

You mean like what was mentioned in post #5?

 

Yes, I see you had already mentioned it.