Question

db is an angle bisector of ∠adc. which statement best describes the relationship between the triangle abd and cbd? triangles abd and cbd are congruent by the sss congruence postulate. triangles abd and cbd are similar by the sss similarity postulate. triangles abd and cbd are congruent by the sas congruence postulate. triangles abd and cbd are similar by the sas similarity postulate.

Explanation:

Step1: Identify given information

We know \( AD = CD = 15 \), \( DB \) is common to both \( \triangle ABD \) and \( \triangle CBD \), and \( DB \) bisects \( \angle ADC \), so \( \angle ADB=\angle CDB \).

Step2: Analyze congruence/similarity

For congruence, SSS (Side - Side - Side) congruence postulate states that if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. Here, \( AD = CD \), \( DB = DB \) (common side), and we need to check the third side. Wait, actually, since \( DB \) is an angle bisector, \( \angle ADB=\angle CDB \), \( AD = CD \), \( DB = DB \). Wait, no, let's re - examine. The sides: \( AD = CD = 15 \), \( DB \) is common, and \( AB \) and \( CB \)? Wait, no, looking at the diagram, \( AB \) and \( CB \): Wait, actually, in \( \triangle ABD \) and \( \triangle CBD \), \( AD = CD \) (given as 15 each), \( DB = DB \) (common side), and \( \angle ADB=\angle CDB \) (since \( DB \) is an angle bisector). Wait, no, SSS congruence would require three sides. Wait, maybe I made a mistake. Wait, the first option: "Triangles ABD and CBD are congruent by the SSS Congruence Postulate". Let's check the sides. \( AD = CD = 15 \), \( DB \) is common, and \( AB = CB \)? Wait, no, from the diagram, \( AB \) and \( CB \): Wait, the line \( AC \) is split by \( B \), so \( AB = BC \)? Wait, maybe the diagram shows that \( B \) is the mid - point? Wait, the diagram has \( A - B - C \) colinear? So \( AB = BC \)? Wait, no, the key is: \( AD = CD = 15 \), \( DB = DB \), and if \( AB = CB \), but actually, the SSS congruence: if \( AD = CD \), \( DB = DB \), and \( AB = CB \), then SSS. But also, since \( DB \) is an angle bisector, \( \angle ADB=\angle CDB \). But the first option says "Triangles ABD and CBD are congruent by the SSS Congruence Postulate". Let's verify the sides: \( AD = CD \) (15 each), \( DB \) is common, and \( AB = CB \) (since \( B \) is on \( AC \), maybe \( AB = BC \)). So three sides: \( AD = CD \), \( DB = DB \), \( AB = CB \), so by SSS congruence postulate, \( \triangle ABD\cong\triangle CBD \).

Step3: Eliminate other options

• Option 2: SAS (Side - Angle - Side) congruence postulate requires two sides and the included angle. But we have three sides equal, so not SAS.
• Option 3: SSS similarity postulate is for similarity, but we are talking about congruence (since sides are equal, not just proportional).
• Option 4: SAS similarity postulate is for similarity, not congruence.

Answer:

Triangles ABD and CBD are congruent by the SSS Congruence Postulate (the first option among the given choices)