Question

given: bd bisects ∠abc, ∠bad ≅ ∠bcd
prove: △abd ≅ △cbd

1. bd bisects ∠abc,
2. ∠abd ≅ ∠cbd
3. ∠bad ≅ ∠bcd
4. bd ≅ bd
5. △abd ≅ △cbd
6. given

2.
3.
4.
5.

Explanation:

Step1: Definition of Angle Bisector

Since \(BD\) bisects \(\angle ABC\), by the definition of an angle bisector, \(\angle ABD\cong\angle CBD\).

Step2: Given

It is given that \(\angle BAD\cong\angle BCD\).

Step3: Reflexive Property

For any segment \(BD\), by the reflexive property of congruence, \(\overline{BD}\cong\overline{BD}\).

Step4: AAS (Angle - Angle - Side)

In \(\triangle ABD\) and \(\triangle CBD\), we have \(\angle ABD\cong\angle CBD\) (from Step 1), \(\angle BAD\cong\angle BCD\) (given), and \(\overline{BD}\cong\overline{BD}\) (Step 3). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle ABD\cong\triangle CBD\).

Answer:

1. REASONS Line 2 - Definition of Angle Bisector
2. REASONS Line 3 - Given
3. REASONS Line 4 - Reflexive Property
4. REASONS Line 5 - AAS