Question 758773
Since given that {{{AD}}} is perpendicular to {{{BC}}} => angle {{{ADC=90}}}degrees and angle {{{ADB=90}}} degrees,
 
so angle {{{ADC}}} is congruent to angle {{{ADB}}}
  
The measure of angle {{{BAD}}} is equal to {{{180}}} minus {{{90}}} minus the measure of angle {{{ADB}}}, 

but since the measure of {{{ADB= ADC}}},
 
the measure of angle {{{BAD}}} is equal to {{{180}}} minus {{{90}}} minus the measure of angle {{{ADC}}}
  
However, that is exactly the measure of angle {{{CAD}}}.
  
Hence angle {{{BAD}}} is congruent to angle {{{CAD}}}.
  
Since {{{AD = AD}}}, triangle {{{ADC}}} is congruent to triangle {{{ADB}}} by {{{SAS}}}. 

 Then {{{AB = AC}}} by {{{CPCT}}}, and finally {{{ABC}}} is {{{isosceles}}} by {{{definition}}}.


or:


Given: AD perpendicular to BC; angle BAD congruent to CAD
Prove: ABC is isosceles


Proof:
statement..........................................reason
1. angle {{{BAD}}} congruent to angle {{{CAD}}} ..........................given
2.{{{AD}}} is perpendicular to {{{BC}}}, => {{{BDA}}} is congruent to the angle {{{CDA}}} ..........all right angles are congruent
3. {{{AD}}} is congruent to {{{AD}}}.......................... reflexive property
4. triangle {{{BAD}}} congruent to triangle {{{CAD}}} ...........by SAS
5. {{{AB}}} is congruent to {{{AC}}} ...................corresponding parts of congruent triangles are congruent
6. triangle {{{ABC}}} is isosceles ....................it has two congruent sides