Question

given: bd bisects ∠abc, ∠bad ≅ ∠bcd. prove: δabd ≅ δcbd. statements: 1. bd bisects ∠abc. 2. ∠bad ≅ ∠bcd. 3. ∠abd ≅ ∠cbd. 4. bd ≅ bd. 5. δabd ≅ δcbd. reasons: 1. given. 2. given. 3. 4. 5.

Explanation:

Step1: Explain statement 3

By definition of angle - bisector, if a ray bisects an angle, it divides the angle into two congruent angles. Since $\overline{BD}$ bisects $\angle ABC$, $\angle ABD\cong\angle CBD$.

Step2: Explain statement 4

The reflexive property of congruence states that any segment is congruent to itself. So, $\overline{BD}\cong\overline{BD}$.

Step3: Explain statement 5

We have $\angle BAD\cong\angle BCD$ (given), $\angle ABD\cong\angle CBD$ (from the angle - bisector property), and $\overline{BD}\cong\overline{BD}$ (reflexive property). By the AAS (Angle - Angle - Side) congruence criterion, which states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, the two triangles are congruent, $\triangle ABD\cong\triangle CBD$.

Answer:

1. Definition of angle - bisector; 4. Reflexive property of congruence; 5. AAS (Angle - Angle - Side) congruence criterion