Question

i) given:
∠1≅∠2 & \\(\overline{ba} \cong \overline{bc}\\)
prove:
\\(\triangle abd \cong \triangle cbd\\)

Explanation:

Step1: Identify Common Side

In triangles \( \triangle ABD \) and \( \triangle CBD \), side \( BD \) is common to both triangles. So, \( \overline{BD} \cong \overline{BD} \) (by the Reflexive Property of Congruence).

Step2: List Known Congruences

We know from the given: \( \angle 1 \cong \angle 2 \), \( \overline{BA} \cong \overline{BC} \), and now \( \overline{BD} \cong \overline{BD} \).

Step3: Apply SAS Congruence

Using the Side - Angle - Side (SAS) Congruence Postulate, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. For \( \triangle ABD \) and \( \triangle CBD \):

• Side: \( \overline{BA} \cong \overline{BC} \) (given)
• Angle: \( \angle 1 \cong \angle 2 \) (given)
• Side: \( \overline{BD} \cong \overline{BD} \) (reflexive property)

So, by SAS, \( \triangle ABD \cong \triangle CBD \).

Answer:

To prove \( \triangle ABD \cong \triangle CBD \):

1. Statement 1: \( \angle 1 \cong \angle 2 \)

• Reason 1: Given

1. Statement 2: \( \overline{BA} \cong \overline{BC} \)

• Reason 2: Given

1. Statement 3: \( \overline{BD} \cong \overline{BD} \)

• Reason 3: Reflexive Property of Congruence (a segment is congruent to itself)

1. Statement 4: \( \triangle ABD \cong \triangle CBD \)

• Reason 4: Side - Angle - Side (SAS) Congruence Postulate (since \( \overline{BA} \cong \overline{BC} \), \( \angle 1 \cong \angle 2 \), and \( \overline{BD} \cong \overline{BD} \))