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

Correct answers:
- AD = BD — True.
- m∠ACD = m∠BCD — True.
- m∠CDA = m∠CDB — True.

Reasoning (brief):
- By the Angle Bisector Theorem, AD/DB = AC/CB. Since AC = CB, AD/DB = 1, so AD = BD.
- CD is given as the bisector of ∠ACB, so ∠ACD = ∠BCD by definition.
- With AC = BC, AD = BD, and CD common, triangles ΔADC and ΔBDC are congruent (SSS). Hence their corresponding angles at D are equal, so ∠CDA = ∠CDB.

Not true:
- AC = CD — not necessarily (no reason the leg equals the bisector length).
- m∠DCA = m∠DAC — not necessarily; that would require AC = AD, which does not follow from the given information.