Question

What are the steps for Given: ∠A≅∠C and BD bisects ∠ABC.
Prove: AC ⊥ BD.

Answer 1

Let A, B, C be the vertices of triangle ABC and let D be the point where BD meets AC (so BD bisects ∠ABC).

Proof (two-column / statement-reason):

1. ∠A ≅ ∠C.
Reason: Given.

2. AB = BC.
Reason: In a triangle, equal base angles imply the opposite sides are equal.

3. ∠ABD ≅ ∠DBC.
Reason: Given (BD bisects ∠ABC).

4. BD = BD.
Reason: Reflexive property.

5. ΔABD ≅ ΔCBD.
Reason: SAS (AB = BC, ∠ABD = ∠DBC, BD = BD).

6. ∠ADB ≅ ∠BDC.
Reason: CPCTC (corresponding parts of congruent triangles).

7. ∠ADB and ∠BDC are a linear pair, so ∠ADB + ∠BDC = 180°.
Reason: A, D, C are collinear (D lies on AC).

8. From 6 and 7, 2·∠ADB = 180°, so ∠ADB = 90°.
Reason: Algebra.

9. Therefore BD ⟂ AC.
Reason: Definition of perpendicular lines (an angle of 90°).

Thus BD is perpendicular to AC.