G.527 there plane angles — Heath

XI.22

If there be three plane angles of which two, taken together in any manner, are greater than the remaining one, and they are contained by equal straight lines, it is possible to construct a triangle out of the straight lines joining the extremities of the equal straight lines.

Let there be three plane angles < A B C ), < D E F ), < G H K ), of which two, taken together in any manner, are greater than the remaining one, namely the angles < A B C ), < D E F ) greater than the angle < G H K ), the angles < D E F ), < G H K ) greater than the angle < A B C ), and, further, the angles < G H K ), < A B C ) greater than the angle < D E F ); let the straight lines / A B /, / B C /, / D E /, / E F /, / G H /, / H K / be equal, and let / A C /, / D F /, / G K / be joined;

I say that it is possible to construct a triangle out of straight lines equal to / A C /, / D F /, / G K /, that is, that any two of the straight lines / A C /, / D F /, / G K / are greater than the remaining one.

Now, if the angles < A B C ), < D E F ), < G H K ) are equal to one another, it is manifest that, / A C /, / D F /, / G K / being equal also, it is possible to construct a triangle out of straight lines equal to / A C /, / D F /, / G K /.

But, if not, let them be unequal, and on the straight line / H K /, and at the point H on it, let the angle < K H L ) be constructed equal to the angle < A B C ); let / H L / be made equal to one of the straight lines / A B /, / B C /, / D E /, / E F /, / G H /, / H K /, and let / K L /, / G L / be joined.

Now, since the two sides / A B /, / B C / are equal to the two sides / K H /, / H L /, and the angle at B is equal to the angle < K H L ),

[G.24]therefore the base / A C / is equal to the base / K L /.

And, since the angles < A B C ), < G H K ) are greater than the angle < D E F ),

while the angle < A B C ) is equal to the angle < K H L ),

therefore the angle < G H L ) is greater than the angle < D E F ).

And, since the two sides / G H /, / H L / are equal to the two sides / D E /, / E F /, and the angle < G H L ) is greater than the angle < D E F ),

[G.49]therefore the base / G L / is greater than the base / D F /.

But / G K /, / K L / are greater than / G L /.

Therefore / G K /, / K L / are much greater than / D F /.

But / K L / is equal to / A C /;

therefore / A C /, / G K / are greater than the remaining straight line / D F /.

Similarly we can prove that / A C /, / D F / are greater than / G K /, and further / D F /, / G K / are greater than / A C /.

Therefore it is possible to construct a triangle out of straight lines equal to / A C /, / D F /, / G K /.

Q. E. D.

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G.528