Question

Triangle A C B is cut by bisector C D. The lengths of sides A C and C B are congruent.
CD bisects ∠ACB. Which statements must be true? Check all that apply.

AD = BD
AC = CD
m∠ACD = m∠BCD
m∠CDA = m∠CDB
m∠DCA = m∠DAC

Answer 1

Since \( AC \) is congruent to \( CB \) (meaning \( AC = CB \)), and \( CD \) is the angle bisector of \( \angle ACB \), we can conclude that several properties hold true based on the properties of triangles and angle bisectors. Let's evaluate each statement:

1. **\( AD = BD \)**: True. Since \( CD \) is the angle bisector of \( \angle ACB \) and \( AC = CB \), by the Angle Bisector Theorem, this implies that \( AD = BD \).

2. **\( AC = CD \)**: False. There’s no information to suggest that the lengths of side \( AC \) and segment \( CD \) are equal.

3. **\( m∠ACD = m∠BCD \)**: True. Because \( CD \) bisects \( \angle ACB \), this means that \( m∠ACD = m∠BCD \).

4. **\( m∠CDA = m∠CDB \)**: False. There’s no guarantee that angles \( CDA \) and \( CDB \) are equal; they depend on the specific triangle form and angle measures.

5. **\( m∠DCA = m∠DAC \)**: False. These angles aren’t necessarily equal based on the given properties.

To summarize, the statements that must be true based on the information given are:
- \( AD = BD \)
- \( m∠ACD = m∠BCD \)

Thus, check those two statements.