Question

the bisector of the angle at the vertex of an isosceles triangle bisects the base at right angles

Answer 1

Yes....................

Answer 2

Given: Isosceles triangle ABC with AB congruent to AC [you draw]
AD bisects angle BAC [you draw ; endpoint D on base BC] 
Prove: AD perpendicular to BC 
1. Isosceles triangle ABC with AB congruent to AC 1. given
AD bisects angle BAC
2. angle BAD congruent to angle CAD 2. def. of angle bisector
3. AD congruent to AD 3. reflexive property


4. triangle BAD congruent to triangle CAD 4. SAS
5. angle BDA congruent to angle CDA 5. CPCTC
6. m angle BDA + m angle CDA = 180 6. linear pair postulate
7. m angle BDA = m angle CDA 7. def. congruent angles
8. m angle BDA + m angle BDA = 180 8. substitution
9. 2*m angle BDA = 180 9. substitution
10. m angle BDA = 90 10. division prop.
11. AD perpendicular to BC 11. def. perpendicular

hope it helps you