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 $overline{ac}$; reasons: given; definition of midpoint; reflexive property of congruence; $\triangle abdcong\triangle cbd$; corresponding parts of congruent triangles are congruent; $overline{ac}congoverline{ac}$, $overline{ad}congoverline{cd}$, $overline{bd}congoverline{bd}$, $angle acongangle c$, $angle abdcongangle cbd$, $angle adbcongangle cdb$; side side side, side angle side, angle angle side

Explanation:

Step1: Use mid - point definition

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

Step2: Apply reflexive property

For side \(\overline{BD}\), by the reflexive property of congruence, \(\overline{BD}\cong\overline{BD}\).

Step3: Prove triangle congruence

We know \(\overline{AB}\cong\overline{CB}\), \(\overline{AD}\cong\overline{CD}\) and \(\overline{BD}\cong\overline{BD}\). So, by the Side - Side - Side (SSS) congruence criterion, \(\triangle ABD\cong\triangle CBD\).

Step4: Use corresponding parts of congruent triangles

Since \(\triangle ABD\cong\triangle CBD\), by the property that corresponding parts of congruent triangles are congruent (CPCTC), \(\angle A\cong\angle C\).

Answer:

Statements	Reasons
\(\overline{AD}\cong\overline{CD}\)	Definition of mid - point
\(\overline{BD}\cong\overline{BD}\)	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