corollary to isosceles triangle theorem

how to prove

1)the measure of each angle of an equilateral triangle is 60.

2)the bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

Thanks.

1)
The angles in any triangle have to add up to 180 degrees, right? 3 x 60 = 180

2)
??

2.
The vertex angle is 60o. If that is bisected, that makes two 30o angles of the vertex angle. And that makes the angle at the base of the triangle a right angle so we have two 30-60-90 degree triangles. The hypotenuse of each triangle is an original side of the original equilateral triangle and the sin 30 = base/hypotenuse. Since the angle is the same and the hypotenuse is the same length, then the length of each part of the base must be the same. I'm sure there is a more esoteric way of proving this but I think this covers the basics.

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  1. i don't know, i just need the definiton of this theorem

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