Question

applying theorems about isosceles triangles
\\(\overline{cd}\\) bisects \\(\angle acb\\). which statements must be true?
check all that apply.
\\(\square\\) \\(ad = bd\\)
\\(\square\\) \\(ac = cd\\)
\\(\square\\) \\(m\angle acd = m\angle bcd\\)
\\(\square\\) \\(m\angle cda = m\angle cdb\\)
\\(\square\\) \\(m\angle dca = m\angle dac\\)

Explanation:

Step1: Use angle bisector definition

By definition, a bisector splits an angle into two congruent angles. Since $\overline{CD}$ bisects $\angle ACB$, we get:
$m\angle ACD = m\angle BCD$

Step2: Analyze isosceles triangle properties

$\triangle ACB$ is isosceles with $AC=BC$. Combined with $\angle ACD=\angle BCD$ and $CD=CD$, by SAS congruence, $\triangle ACD \cong \triangle BCD$. Corresponding parts of congruent triangles are equal, so:
$AD=BD$ and $m\angle CDA = m\angle CDB$

Step3: Eliminate incorrect statements

There is no given or deducible information that $AC=CD$, nor that $m\angle DCA = m\angle DAC$.

Answer:

• AD = BD
• $m\angle ACD = m\angle BCD$
• $m\angle CDA = m\angle CDB$