Question

given that ∠abc ≅ ∠dbe, which statement must be true?
∠abc ≅ ∠abd
∠abd ≅ ∠cbe
∠cbd ≅ ∠dbe
∠cbd ≅ ∠abc

Explanation:

Step1: Use the Angle - Angle relationship

Since \(\angle ABC\cong\angle DBE\), we know that \(\angle ABC=\angle DBE\).
Let \(\angle ABC = \angle1\), \(\angle ABD=\angle2\), \(\angle CBD=\angle3\), \(\angle CBE=\angle4\). So \(\angle1=\angle2 + \angle3\) and \(\angle4=\angle3+\angle DBE\). But \(\angle1=\angle DBE\).
If \(\angle ABC\cong\angle DBE\), then \(\angle ABC-\angle CBD=\angle DBE-\angle CBD\) (by the subtraction property of equality for angles).

Step2: Simplify the equation

\(\angle ABC-\angle CBD=\angle ABD\) and \(\angle DBE - \angle CBD=\angle CBE\). Since \(\angle ABC=\angle DBE\), we get \(\angle ABD=\angle CBE\), so \(\angle ABD\cong\angle CBE\).

Answer:

\(\angle ABD\cong\angle CBE\) (the second option)