Question

The bisectors of the exterior angles at B, C of the triangle ABC meet at Y. Prove that angle BYC is less than a right angle.

Answer 1

@livelaughlilz Please post a sketch of the diagram that depicts the information given in your post. Thanks.

Answer 2

@Directrix there was no diagram given

Answer 3

Right. Interpret the words and draw the diagram.

Answer 4

(we were taught that an exterior angle is one that is both adjacent and supplementary to it's interior angle)

Answer 5

That is correct. The sum of all 3 exterior angles of a triangle is 360. Also, the measure of one exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. So, we have to figure out what to put together to show what we want to prove.

The diagram looks good.

Answer 6

oh. um, we haven't yet learned that the sum of the measures of the exterior angles in a triangle is 360. We also haven't learned that the measure of an exterior angle is the sum of the remote interior angles, although we did learn that it is greater than any of the two remote interior angles. do you know how to prove it without using these theorems? the proof should focus on inequalities, since that is what was just covered in the textbook pages before this problem. if you don't know how to do it without those theorems, i would still like that solution

Answer 7

I don't think we have to use theorems you have not studied. I was just writing down theorems related to exterior angles of a triangle thinking that some idea for a proof technique might pop into mind.

I'm wondering about the diagram now. There are two exterior angles per vertex. Of course, we just want to deal with one of them but the two exterior angles must be in the same group. That is not going to make much sense just yet.

Answer 8

Right, I was thinking about that too. I was also wondering whether this should be a direct or indirect proof. All I really seem to have is a few angle congruencies.

Answer 9

Here are the two sets. I am thinking you drew one from one set and the second from the other. I may be wrong.

Answer 10

Oh I think you're right. So the diagram IS wrong?

Answer 11

What if we use this diagram? What do you think?