P l a n e G e o m e t r y
An Adventure in Language and Logic
THE THREE ANGLES OF A TRIANGLE
Book I. Propositions 31 and 32
WE HAVE SEEN THE fundamental theorems of parallel lines. They are Propostion 27 and its converse, Proposition 29. Here again is
Proposition 29. If two straight lines are parallel, then a straight line that meets them makes the alternate angles equal; it makes the exterior angle equal to the opposite interior angle on the same side; and it makes the interior angles on the same side equal to two right angles.
Whenever we know that two straight lines are parallel, those are the conclusions that we can make.
We will now present the well-known theorem about the three angles of a triangle, which is that they are equal to two right angles. But first, we must solve the following problem.
PROPOSITION 31. PROBLEM
The construction and proof are straightforward, and they are left as an exercise. (Problem 1)
We are now ready to prove that the three angles of a triangle are together equal to two right angles.
For, on extending BC to D, and drawing CE parallel to AB, the angles BAC, ACE labeled 2 are alternate angles; angle 3 on the right is exterior to the parallel lines and is equal to the interior angle 3. Angles 1, 2, 3 along BD form a straight angle composed of two right angles. And they are equal to angles 1, 2, 3 of the triangle!
Here is the formal proposition.
PROPOSITION 32. THEOREM
Corollary. All the interior angles of any polygon are together equal to twice as many right angles as the figure has sides, less four right angles.
For, we can divide any polygon ABCDE into as many triangles as the figure has sides, by drawing straight lines from any point F within the figure to each vertex.
And those angles are all the interior angles of the figure, together with all the angles at the point F,
Therefore all the interior angles of any polygon are equal to twice as many right angles as the figure has sides, less four right angles.
Please "turn" the page and do some Problems.
Continue on to the next proposition.
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Copyright © 2006-2007 Lawrence Spector
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