Question

dc || ab
segment ad is congruent to segment bd because they are the same segment. angles a and c are given to be congruent because they are both
ad
cd
db
since line dc and line ab are given to be , angle dba is congruent to angle bdc by
angle cbcd congruent to angle adb by theorem.
the theorem.
therefore, triangle abd is congruent to triangle cdb by theorem.

Explanation:

Step1: Segment Congruence

Segment \( DB \) is congruent to segment \( DB \) because they are the same segment (reflexive property).

Step2: Parallel Lines Property

Since line \( DC \) and line \( AB \) are given to be parallel (\( DC\parallel AB \)), angle \( DBA \) is congruent to angle \( BDC \) by the alternate - interior angles theorem.

Step3: Right - Angle Congruence

Angles \( A \) and \( C \) are given to be congruent (both right angles, \( \angle A=\angle C = 90^{\circ}\)). Angle \( CBD \) is congruent to angle \( ADB \) by the alternate - interior angles theorem.

Step4: Congruence Theorem

By the AAS (Angle - Angle - Side) Theorem, triangle \( ABD \) is congruent to triangle \( CDB \).

Answer:

1. Segment \( DB \) is congruent to segment \( DB \) (reflexive property).
2. \( DC\parallel AB \), alternate - interior angles theorem.
3. Alternate - interior angles theorem.
4. AAS (Angle - Angle - Side) Theorem.