Decipher Triangle Theorem in David Wells Book

Question

It's now about the tenth time I've tried to read David Wells's book "You are a Mathematician" and not got past the first chapter because I can't understand his writing.

There is a theorem about triangles which I'm sure is very simple expressed in these two images:

https://i.stack.imgur.com/hqaUQ.jpg

https://imgur.com/a/3hrE85m

However, the relationship between the first diagram and the second is not made clear (1.4 and 1.6), and there are no labels on the second one to help with the explanation.

He seems to be saying that in a triangle cut into two parts by a line from one vertex to the opposite side, if the angle sums of the three triangles thus formed are equal then the angle sums of the smaller triangles must be equal to the sum of the angles on either side of the point where the line form the vertex meets the opposite side.

However I see no argument explaining why this should be so.

Could someone please provide a clearer explanation, perhaps with the aid of a diagram?

Also could someone please give me an idea of how clear they consider the explanation given in the book to be? I'm curious to know if my confusion is "normal."

Answer 1

Suppose the sum of the angles of any triangle is some constant $\pi$. The sum of the angles of the large triangle is the sum of the angles of the two small triangles, minus $X$ and $Y$. Hence: $$ \pi=\pi+\pi-X-Y, \quad\text{that is:}\quad X+Y=\pi. $$

Answer 2

The overall argument from Fig. 1.4 and Fig. 1.6 appears to be as you suggest, and the argument from Fig. 1.6 in particular appears to be that if, as conjectured, the angle sum of any triangle has a constant value $S$, then, in the figure below: \begin{align*} 2S & = (\text{angle sum of } \triangle ABD) + (\text{angle sum of } \triangle ADC) \\ & = (\alpha_2 + \beta + X) + (\alpha_1 + Y + \gamma) \\ & = ((\alpha_1 + \alpha_2) + \beta + \gamma) + (X + Y) \\ & = (\text{angle sum of } \triangle ABC) + (X + Y) \\ & = S + (X + Y), \end{align*} whence $S = X + Y$. I don't know why he expresses this conclusion as [emphasis mine] "the angle sum of each of the smaller triangles must be $X + Y$": that does seem rather confusing. Perhaps it is intended to express the fact that $S$ is subtracted from both sides of the equation, leaving the angle sum of one small triangle?

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I forgot to address an earlier part of the question:

However, the relationship between the first diagram and the second is not made clear (1.4 and 1.6), and there are no labels on the second one to help with the explanation.

I rather agree. I think Wells's argument from Fig. 1.4 is that when a line rotates, the angle it makes with any other (fixed) line changes by the same amount, viz. the angle of the rotation. Rather more clearly, he concludes from this that the angle sum of $\triangle ABC$ in Fig. 1.4 [no relation to $\triangle ABC$ in my annotated version of Fig. 1.6 - I'm sorry about the clash of notation] is equal to the angle sum of $\triangle AB'C'$.

I must admit that I don't at all see how he concludes from this that the angle sums of any two triangles are equal (or even likely to be equal). Perhaps I'm missing something obvious, but I'm inclined to agree that his style of argument - in this passage at least - is obscure. Most likely, his hidden argument is that any two triangles (regarding similar triangles as equal) can be obtained from one another by a series of such rotations (in fact, by at most two such rotations); and the angle sum is preserved at each stage, therefore it is preserved from beginning to end; but I think he should have stated this explicitly. (Of course, I may be misinterpreting his intention.)