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", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "SAS Congruence Postulate", "Reflexive Property of Congruence" ], "new_concepts": [ "Angle Bisector Definition", "Isosceles Triangle Definition" ], "current_concepts": [ "SAS Congruence Postulate", "Reflexive Property of Congruence", "Angle Bisector Definition", "Isosceles Triangle Definition" ] } ## Identify given congruent parts Using the *Reflexive Property of Congruence* knowledge point $$ \overline{CB} \cong \overline{CB} $$ Using the definition of an angle bisector, since $\overline{CB}$ bisects $\angle ACD$: $$ \angle ACB \cong \angle DCB $$ ## Determine required SAS components Using the *SAS Congruence Postulate* knowledge point To prove $\Delta ABC \cong \Delta DBC$ using SAS, we need: 1. Side: $\overline{CB} \cong \overline{CB}$ (already established) 2. Included Angle: $\angle ACB \cong \angle DCB$ (already established) 3. Side: $\overline{AC} \cong \overline{DC}$ Thus, any information that proves $\overline{AC} \cong \overline{DC}$ (or $AC = DC = 52\text{ cm}$) is a correct option. ## Evaluate the given options - Option 1: $m\angle ABC = 125^\circ$ and $\overline{AB} \cong \overline{DB}$. This provides SSA, which does not prove congruence. - Option 2: $\Delta ACD$ is isosceles with base $\overline{AD}$. By definition, this means $\overline{AC} \cong \overline{DC}$, which is correct. - Option 3: $\Delta ABD$ is isosceles with base $\overline{AD}$. This means $\overline{AB} \cong \overline{DB}$, which does not help prove SAS for $\Delta ABC \cong \Delta DBC$. - Option 4: $CD = 52\text{ cm}$. Since $AC = 52\text{ cm}$, this means $AC = CD$, so $\overline{AC} \cong \overline{DC}$, which is correct. - Option 5: $AB = 29\text{ cm}$. Since $BD = 29\text{ cm}$, this means $AB = BD$, so $\overline{AB} \cong \overline{DB}$. Along with $m\angle ABC = 125^\circ$ (which is equal to $m\angle DBC$ by symmetry/reflection), this gives SAS using sides $\overline{CB}$, $\overline{AB}$ and included angle $\angle ABC$. Thus, this is also correct.

$m\angle ABC = 125^\circ$ 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 given congruent parts

Determine required SAS components

Evaluate the given options

Answer: