Geometric proof demonstrating triangle congruence using ASA postulate.

Let’s go step by step to fill in the missing reason for Step 3:
Step 3: ∠EAC ≅ ∠BAD — Reason: ?

We are told in Step 1 that segment ED bisects segment BC. That means point A (where ED and BC cross) is the midpoint of BC, so AB ≅ AC (which is Step 2).

Also, from the diagram (even though we’re not describing it), we can see that lines ED and BC intersect at point A, forming vertical angles.

Look at angles ∠EAC and ∠BAD. These two angles are vertical angles — they are opposite each other where two lines cross (lines ED and BC intersect at A).

And here’s a key geometry fact:
> Vertical angles are always congruent.

So, ∠EAC and ∠BAD are vertical angles → therefore, they are congruent.

That’s why Step 3 says ∠EAC ∠BAD — the reason is Vertical Angles Theorem.

Let’s double-check the rest of the proof to make sure this fits:

- Step 4: ∠ABD ≅ ∠ACE — given as “Alternate Interior Angles” — which makes sense if EC || BD (given in Step 1), and line BC is the transversal.
- Step 5: Triangles ABD and ACE are congruent by ASA — because we have:
- ∠ABD ≅ ∠ACE (Step 4)
- AB ≅ AC (Step 2)
- ∠BAD ≅ ∠CAE (Step 3 — vertical angles)

Wait — hold on! In Step 3, it says ∠EAC ∠BAD. But in triangle ACE, the angle at A is ∠CAE (same as ∠EAC). And in triangle ABD, the angle at A is ∠BAD. So yes, those are the included angles between the sides and the other angles.

Actually, let’s be precise with naming:

In triangle ABD: vertices A, B, D → angles at A, B, D.

In triangle ACE: vertices A, C, E → angles at A, C, E.

For ASA congruence, we need two angles and the included side.

From Step 4: ∠ABD ≅ ∠ACE → that’s angle at B in first triangle, angle at C in second.

From Step 2: AB ≅ AC → that’s the side between angle at A and angle at B in first triangle; and between angle at A and angle at C in second triangle.

Then Step 3: ∠BAD ≅ ∠CAE → that’s the angle at A in both triangles.

Yes — so ASA works: Angle (∠ABD) - Side (AB) - Angle (∠BAD) in triangle ABD corresponds to Angle (∠ACE) - Side (AC) - Angle (∠CAE) in triangle ACE.

Perfect.

So the missing reason for Step 3 is definitely: Vertical Angles Theorem

Sometimes textbooks say “Vertical angles are congruent” or just “Vertical Angles”.

But the standard reason name is: Vertical Angles Theorem

Final Answer: Vertical Angles Theorem