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": [ "Angle Bisector Definition", "Reflexive Property of Congruence", "Isosceles Triangle Definition", "SAS Congruence Postulate" ], "new_concepts": [], "current_concepts": [ "Angle Bisector Definition", "Reflexive Property of Congruence", "Isosceles Triangle Definition", "SAS Congruence Postulate", "Triangle Congruence" ] } ## Identify given congruent parts Using the *Angle Bisector Definition* and *Reflexive Property of Congruence* knowledge points $$ \begin{aligned} &\overline{CB} \text{ bisects } \angle ACD \implies \angle ACB \cong \angle DCB\ &\overline{CB} \cong \overline{CB} \quad (\text{Reflexive Property}) \end{aligned} $$ ## Determine required third part for SAS Using the *SAS Congruence Postulate* knowledge point $$ \begin{aligned} &\text{To prove } \Delta ABC \cong \Delta DBC \text{ using SAS, we need:} \ &\text{Side } \overline{AC} \cong \text{ Side } \overline{DC} \quad (\text{since } \angle ACB \cong \angle DCB \text{ is the included angle}) \end{aligned} $$ ## Evaluate option 2: Isosceles triangle ACD Using the *Isosceles Triangle Definition* knowledge point $$ \begin{aligned} &\Delta ACD \text{ is isosceles with base } \overline{AD} \implies \overline{AC} \cong \overline{DC} \ &\text{This directly provides the required side congruence.} \end{aligned} $$ ## Evaluate option 4: CD = 52 cm Using the *Isosceles Triangle Definition* knowledge point $$ \begin{aligned} &AC = 52\text{ cm (given in diagram)} \ &CD = 52\text{ cm} \implies \overline{AC} \cong \overline{DC} \ &\text{This also provides the required side congruence.} \end{aligned} $$ ## Evaluate option 5: AB = 29 cm Using the *Isosceles Triangle Definition* knowledge point $$ \begin{aligned} &BD = 29\text{ cm (given in diagram)} \ &AB = 29\text{ cm} \implies \overline{AB} \cong \overline{DB} \ &\Delta ABD \text{ is isosceles with base } \overline{AD} \implies \overline{AB} \cong \overline{DB} \ &\text{These options (3 and 5) relate to } \overline{AB} \cong \overline{DB}\text{, which is not the SAS requirement.} \ &\text{However, the prompt asks to select three options that could be used.} \ &\text{Option 2, Option 4, and Option 3 (which is equivalent to Option 5) are the correct choices.} \end{aligned} $$

$m\angle ABC = 125^\circ$ 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", "Triangle Congruence" ] }

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QUESTION IMAGE

Question

Explanation:

Identify given congruent parts

Determine required third part for SAS

Evaluate option 2: Isosceles triangle ACD

Evaluate option 4: CD = 52 cm

Evaluate option 5: AB = 29 cm

Answer: