This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To determine the congruence between the triangles, we compare their side lengths. Given information: For ABC: AB = 5 cm AC = 8 cm BC = 4 cm For DBC: BD = 8 cm CD = 5 cm BC = 4 cm (This side is common to both triangles) Step 1: Compare the corresponding sides. Let's list the sides of ABC and try to match them with the sides of DCB. Side AB = 5 cm in ABC. Side DC = 5 cm in DCB. So, AB = DC. Side BC = 4 cm in ABC. Side CB = 4 cm in DCB. So, BC = CB. Side AC = 8 cm in ABC. Side DB = 8 cm in DCB. So, AC = DB. Step 2: Determine the congruence criterion. Since all three corresponding sides of ABC are equal to the three corresponding sides of DCB, the triangles are congruent by the SSS (Side-Side-Side) congruence criterion. Step 3: Write the correct congruence statement. To write the congruence statement, we need to match the vertices correctly based on the equal sides: Since AB = DC, vertex A corresponds to vertex D, and vertex B corresponds to vertex C. Since BC = CB, vertex B corresponds to vertex C, and vertex C corresponds to vertex B. Since AC = DB, vertex A corresponds to vertex D, and vertex C corresponds to vertex B. Combining these, we get the correspondence: A D B C C B Therefore, the correct congruence statement is ABC DCB. Step 4: Check the given options. (1) ABC DBC, by SAS - Incorrect vertex correspondence and criterion. (2) ABC DCB, by SAS - Incorrect criterion. (3) ABC DBC, by SSS - Incorrect vertex correspondence. (4) ABC DCB, by SSS - Correct vertex correspondence and criterion. The final answer is 4.