Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/202

178 Again, because the straight line BAE meets the parallels AD, EC, the exterior angle BAD is equal to the interior and opposite angle AEC; [I.29.

but the angle BAD has been shewn equal to the angle ACE;

therefore the angle ACE is equal to the angle AEC; [Axiom 1.

and therefore AC is equal to AE. [I. 6.

And, because AD is parallel to EC, [Constr.

one of the sides of the triangle BCE,

therefore BD is to DC as BA is to AE; [VI. 2.

but AE is equal to AC;

therefore BD is to DC as BA is to AC. [V.7.

Next, let BD be to DC as BA is to AC, and join AD: the angle BAC shall be bisected by the straight line AD,

For, let the same construction be made. Then BD is to DC as BA is to AC; [Hypothesis.

and BD is to DC as BA is to AE, [VI. 2.

because AD is, parallel to EC; [Construction.

therefore BA is to AC as BA is to AE; [V. 11.

therefore AC is equal to AE; [V. 9.

and therefore the angle AEC is equal to the angle ACE. [1.5.

But the angle AEC is equal to the exterior angle BAD; [1. 29.

and the angle ACE is equal to the alternate angle CAD; [1. 29.

therefore the angle BAD is equal to the angle CAD;[Ax.l.

that is, the angle BAC is, bisected by the straight line AD.

Wherefore, if the vertical angle &c.