Question

what must be the final reason for a proof, given the information below?

given: ( ac = bd ) and ( angle acb = angle dbc )
prove: ( \triangle abc cong \triangle dcb )

✅ sss
✅ hl
✅ aaa
✅ asa
✅ ssa
✅ sas
✅ aas

Explanation:

• Identify the sides and angles: We are given \(AC = BD\) (a side), \(\angle ACB=\angle DBC\) (an angle). Also, \(BC\) is a common side between \(\triangle ABC\) and \(\triangle DCB\).
• Recall the congruence postulates:
• SSS (Side - Side - Side): Requires three pairs of equal sides. We only know two pairs of sides (\(AC = BD\) and \(BC=CB\)) initially.
• HL (Hypotenuse - Leg): Used for right - angled triangles. There is no information about right angles in the problem.
• AAA (Angle - Angle - Angle): Only gives similarity, not congruence.
• ASA (Angle - Side - Angle): The side must be between the two angles. Here, the given side \(BC\) is between \(\angle ACB\) and \(\angle ABC\) (not directly given in the order for ASA from the problem's given information).
• SSA (Side - Side - Angle): Is not a valid congruence postulate in general.
• SAS (Side - Angle - Side): We have \(AC = BD\) (side), \(\angle ACB=\angle DBC\) (angle), and \(BC = CB\) (side). The angle is between the two sides.
• AAS (Angle - Angle - Side): Requires two angles and a non - included side. We don't have two angles given initially.

Answer:

SAS