[Illustration] To ascertain the height of a tree or of the school building, fold a piece of paper so as to make an angle of 45 deg.. Then walk back from the tree until the top is seen at an angle of 45 deg. with the ground (being therefore careful to have the base of the triangle level). Then the height _AC_ will equal the base _AB_, since _ABC_ is isosceles. A paper protractor may be used for the same purpose. Distances can easily be measured by constructing a large equilateral triangle of heavy pasteboard, and standing pins at the vertices for the purpose of sighting. [Illustration] To measure _PC_, stand at some convenient point _A_ and sight along _APC_ and also along _AB_. Then walk along _AB_ until a point _B_ is reached from which _BC_ makes with _BA_ an angle of the triangle (60 deg.). Then _AC_ = _AB_, and since _AP_ can be measured, we can find _PC_. Another simple method of measuring a distance _AC_ across a stream is shown in this figure. [Illustration] Measure the angle _CAX_, either in degrees, with a protractor, or by sighting along a piece of paper and marking down the angle. Then go along _XA_ produced until a point _B_ is reached from which _BC_ makes with _A_ an angle equal to half of angle _CAX_. Then it is easily shown that _AB_ = _AC_. A navigator uses the same principle when he "doubles the angle on the bow" to find his distance from a lighthouse or other object. [Illustration] If he is sailing on the course _ABC_ and notes a lighthouse _L_ when he is at _A_, and takes the angle _A_, and if he notices when the angle that the lighthouse makes with his course is just twice the angle noted at _A_, then _BL_ = _AB_. He has _AB_ from his log (an instrument that tells how far a ship goes in a given time), so he knows _BL_. He has "doubled the angle on the bow" to get this distance. It would have been possible for Thales, if he knew this proposition, to have measured the distance of the ship at sea by some such device as this: [Illustration] Make a large isosceles triangle out of wood, and, standing at _T_, sight to the ship and along the shore on a line _TA_, using the vertical angle of the triangle. Then go along _TA_ until a point _P_ is reached, from which _T_ and _S_ can be seen along the sides of a base angle of the triangle. Then _TP_ = _TS_. By measuring _TB_, _BS_ can then be found. THEOREM. _The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side_.