Question

ΔABC is an equilateral triangle with AD perpendicular to BC. Prove that ΔADB ≅ ΔADC.

Answer 1

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.