Given $\overline{AC} \cong \overline{BD}$ and $\angle CAB \cong \angle DBA$, prove $\triangle ABC \cong \triangle BAD$.
Understand the Problem
The question provides two statements: $\overline{AC} \cong \overline{BD}$ and $\angle CAB \cong \angle DBA$. The user needs to prove that $\triangle ABC \cong \triangle BAD$. This likely involves using congruence postulates or theorems, such as Side-Angle-Side (SAS), Side-Side-Side (SSS), or Angle-Side-Angle (ASA).
Answer
$\overline{AB} \cong \overline{BA}$ by the Reflexive Property of Congruence, and $\triangle ABC \cong \triangle BAD$ by SAS.
Answer for screen readers
| Statement | Reason |
|---|---|
| 1. $\overline{AC} \cong \overline{BD}$, $\angle CAB \cong \angle DBA$ | Given |
| 2. $\overline{AB} \cong \overline{BA}$ | Reflexive Property of Congruence |
| 3. $\triangle ABC \cong \triangle BAD$ | SAS Congruence Postulate |
Steps to Solve
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State the given information We are given that $\overline{AC} \cong \overline{BD}$ and $\angle CAB \cong \angle DBA$. This is our starting point.
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Identify the shared side Both triangles $\triangle ABC$ and $\triangle BAD$ share the side $\overline{AB}$.
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State the shared side congruence Since $\overline{AB}$ is shared by both triangles, we can say $\overline{AB} \cong \overline{BA}$ by the reflexive property of congruence.
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Apply the SAS Congruence Postulate We have $\overline{AC} \cong \overline{BD}$ (Side), $\angle CAB \cong \angle DBA$ (Angle), and $\overline{AB} \cong \overline{BA}$ (Side). Therefore, by the Side-Angle-Side (SAS) congruence postulate, we can conclude that $\triangle ABC \cong \triangle BAD$.
| Statement | Reason |
|---|---|
| 1. $\overline{AC} \cong \overline{BD}$, $\angle CAB \cong \angle DBA$ | Given |
| 2. $\overline{AB} \cong \overline{BA}$ | Reflexive Property of Congruence |
| 3. $\triangle ABC \cong \triangle BAD$ | SAS Congruence Postulate |
More Information
The Side-Angle-Side (SAS) Congruence Postulate states that if two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the two triangles are congruent.
Tips
A common mistake would be failing to recognize the shared side $\overline{AB}$, or incorrectly applying the congruence postulates.
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