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WE draw from this Proposition several very useful conclusions. The First, that from a given Point there cannot be drawn no more than one Perpen∣dicular to the same Line. Example, Let the Line AB be Perpendicular to the Line BC: I say AC shall not be Perpen∣dicular thereto, because the Right Angle ABC, shall be greater than the Interiour
ACB: Therefore ACB shall not be a Right Angle, neither AC a Perpendi∣cular.
Secondly, that from the same Point A, there can only be drawn two equal Lines; for Example, AC, AD, and that if you draw a Third AE, it shall not be equal. For since AC, AD, are equal, the Angles ACD, ADC, are equal, (by the 5th.) now in the Triangle AEC, the Exteriour Angle ACB is greater than the Interiour AEC; and thus is the Angle ADE greater than AED. Therefore the Line AE, greater than AD; and by consequence AC, AE, are not equal.
Thirdly, if the Line AC maketh the Angle ACB acute, and ACF obtuse, the Perpendicular drawn from A, shall fall on the same Side which the Acute Angle is of; for if one should say that AE is a Perpendicular, and the Angle AEF is Right; then the Right Angle AEF would be greater than the Obtuse Angle ACE. Those conclusions we make use of to measure all Parallelograms, Tri∣angles, and Trapeziams, and to Reduce them into Rectangular Figures.
Use 16.