Step-by-step explanation:
An equilateral triangle means all sides are congruent.
AB≅BC≅AC
All angles are also congruent.
∠ABC≅∠BCA≅∠CAB
AD being perpendicular makes it a bisector of both BC and ∠CAB.
BD≅CD
∠BAD≅∠CAD
Now, there are multiple ways to prove that ΔADB≅ΔADC. All 3 sides and all 3 angles of both triangles are congruent, so you could do it however you want.
1) ASA (one of the possible ASA combinations)
- ∠ABD ≅ ∠ACD because this is an equilateral triangle
- BD ≅ CD because AD bisects BC
- ∠BDA ≅ ∠CDA because AD is perpendicular to BC, both 90°
2) HL (again, one of the multiple possible HL combinations)
- AD is perpendicular to BC, creating 2 right triangles
- AB ≅ AC because ΔABC is equilateral
- AD ≅ DA by the reflexive property, it is congruent to itself
There are many more but I won't write them all out.
