Question

AD perpendicular BC. Dis the midpoint of BC
Using SAS Congruence rule show that

triangle ABD
is congruent to triangle ACD

Is AB = AC? Why?
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Answer 1

In ∆ ABD and ∆ ACD

AD = AD ( common side)

BD = CD ( D is mid point of BC)

< ADB = < ADC = 90° ( AD is perpendicular to BC)

thus ∆ ADB is congruent to ∆ ADC by SAS congruency.

since, the two triangles are congruent, all six parts of one triangle will be equal to the corresponding parts of the other triangle. We have already information about three equal parts which we used to prove the triangles congruent, hence the remaining three parts will also be equal. this is where congruency theorems are useful as it proves two triangles congruent by comparing only three corresponding parts of two triangles

one such part is

side AB in ∆ ADB = side AC in ∆ ADC

so, AB = AC

(you can think which two remaining parts will be equal, a triangle has six parts-- three sides and three angles. I am not taking into account the three vertices)