if $\overline{cb}$ bisects $\angle acd$, what additional information could be used to prove $\delta abc \cong \delta dbc$ using sas? select three options.

$m\angle abc = 125^\circ$ and $\overline{ab} \cong \overline{db}$
$\delta acd$ is isosceles with base $\overline{ad}$
$\delta abd$ is isosceles with base $\overline{ad}$
$cd = 52\text{ cm}$
$ab = 29\text{ cm}$

Question

if $\overline{cb}$ bisects $\angle acd$, what additional information could be used to prove $\delta abc \cong \delta dbc$ using sas? select three options.

$m\angle abc = 125^\circ$ and $\overline{ab} \cong \overline{db}$
$\delta acd$ is isosceles with base $\overline{ad}$
$\delta abd$ is isosceles with base $\overline{ad}$
$cd = 52\text{ cm}$
$ab = 29\text{ cm}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "SAS Congruence Postulate" ], "new_concepts": [], "current_concepts": [ "SAS Congruence Postulate", "Triangle Congruence", "Geometric Proofs" ] } ## Identify existing congruent parts $$ \begin{aligned} &\overline{CB} \cong \overline{CB} \quad \text{(Reflexive Property)} \ &\overline{CB} \text{ bisects } \angle ACD \implies \angle ACB \cong \angle DCB \end{aligned} $$ ## Determine required third part for SAS $$ \begin{aligned} &\text{To prove } \Delta ABC \cong \Delta DBC \text{ using SAS with } \overline{CB} \cong \overline{CB} \text{ and } \angle ACB \cong \angle DCB: \ &\text{We need the other adjacent side of the angle: } \overline{AC} \cong \overline{DC} \end{aligned} $$ ## Evaluate the given options for equivalence to $\overline{AC} \cong \overline{DC}$ $$ \begin{aligned} &1. \text{ Option 2: } \Delta ACD \text{ is isosceles with base } \overline{AD} \implies \overline{AC} \cong \overline{DC} \ &2. \text{ Option 4: } CD = 52\text{ cm} \implies \overline{AC} \cong \overline{DC} \text{ (since } AC = 52\text{ cm)} \ &3. \text{ Option 5: } AB = 29\text{ cm} \implies \overline{AB} \cong \overline{DB} \text{ (since } BD = 29\text{ cm)} \implies \Delta ABC \cong \Delta DBC \text{ by SAS (using } \angle ABC \cong \angle DBC = 125^\circ \text{ and shared } \overline{CB}\text{)} \end{aligned} $$

$m\angle ABC = 125^\circ$ and $\overline{AB} \cong \overline{DB}$

$\Delta ACD$ is isosceles with base $\overline{AD}$

$\Delta ABD$ is isosceles with base $\overline{AD}$

$CD = 52\text{ cm}$

$AB = 29\text{ cm}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "SAS Congruence Postulate" ] }

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QUESTION IMAGE

Question

Explanation:

Identify existing congruent parts

Determine required third part for SAS

Evaluate the given options for equivalence to \(\overline{AC} \cong \overline{DC}\)

Answer: