94
HENRICI.
[Act II.
| Euclid. | Henrici. |
The greater side of a Triangle is opposite to the greater angle.
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| Let ABC be a triangle having AC > AB: then shall the angle ABC be > the angle C.
From AC cut off AD equal to AB; and draw BD. Then, ∵ AB = AD, ∴ the angle ABD = the angle ADB; but the angle ADB is exterior to the Triangle BCD, and ∴ < the angle C; ∴ the angle ABD also > the angle C; much more is the angle ABC > the angle C. Q. E. D. |
Let us now suppose a Triangle ABC, in which the bisector of the angle BAC is not an axis of symmetry. Then the contra-positive form of the theorem of § 162 tells us that AB is not equal to AC, that the angle B is not equal to the angle C, and that the bisector AD of the angle A is not perpendicular to BC, and hence, that the two angles ADB and ADC are unequal. Between these angles there exists the relation
'the angle ABC + the angle BDA = the angle C + the angle CDA,' for each sum makes with half the angle A an angle of continuation. Hence it follows that, if the angle ABC > the angle C, the angle BDA < the angle CDA. |
