Use the definition of an angle bisector. Notice you have two pairs of corresponding angles that are congruent. Are the corresponding included sides congruent?
Statements
Reasons
1.
CB bisects ∠ ABD and ∠ ACD
1.
Given
2.
∠ ABC ≅ ∠ DBC
2.
Definition of angle bisector
3.
BC ≅ BC
3.
Reflexive Property of Congruent Segments
4.
∠ ACB ≅ ∠ DCB
4.
Definition of angle bisector
5.
△ ABC ≅ △ DBC
5.
Angle-Side-Angle (ASA) Congruence Postulate
Practice makes perfect
By the definition of an angle bisector, we have that ∠ ABC ≅ ∠ DBC and ∠ ACB ≅ ∠ DCB. Let's mark them in the given diagram.
Notice that BC is common for both △ ABC and △ DBC, and by the Reflexive Property of Congruent Segments we have BC ≅ BC.
cc
∠ ABC ≅ ∠ DBC & Angle
BC ≅ BC & Included Side
∠ ACB ≅ ∠ DCB & Angle
By the Angle-Side-Angle (ASA) Congruence Postulate we conclude that △ ABC ≅ △ DBC. We summarize this proof in the following two column table.
