Question

in triangle abc. $overline{ab}congoverline{cb}$ and d is the mid - point of $overline{ac}$. prove that $angle acongangle c$. move the options to the spaces to complete the proof.
statements: $overline{ab}congoverline{cb}$ and d is the mid - point of $overline{ac}$, _, _, $\triangle abdcong\triangle cbd$, _
reasons: given, definition of midpoint, reflexive property of congruence, _, corresponding parts of congruent triangles are congruent.
options: $overline{ac}congoverline{ac}$, $overline{ad}congoverline{cd}$, $overline{bd}congoverline{bd}$, $angle acongangle c$, $angle abdcongangle cbd$, $angle adbcongangle cdb$, side side side, side angle side, angle angle side

Explanation:

Step1: Define mid - point

Since $D$ is mid - point of $\overline{AC}$, by definition $\overline{AD}\cong\overline{CD}$.

Step2: Apply reflexive property

For $\overline{BD}$, it is congruent to itself, so $\overline{BD}\cong\overline{BD}$.

Step3: Prove triangle congruence

We have $\overline{AB}\cong\overline{CB}$, $\overline{AD}\cong\overline{CD}$ and $\overline{BD}\cong\overline{BD}$, so $\triangle ABD\cong\triangle CBD$ by SSS.

Step4: Use CPCTC

Since $\triangle ABD\cong\triangle CBD$, corresponding angles $\angle A$ and $\angle C$ are congruent by CPCTC.

Answer:

Statements	Reasons
$\overline{AD}\cong\overline{CD}$	Definition of midpoint
$\overline{BD}\cong\overline{BD}$	Reflexive property of congruence
$\triangle ABD\cong\triangle CBD$	Side - Side - Side (SSS)
$\angle A\cong\angle C$	Corresponding parts of congruent triangles are congruent