Question
$\triangle\text{ABC}$ and $\triangle\text{ABD}$ are on a common base $AB$, and $\ce{AC = BD}$ and $\ce{BC = AD}$ as shown in Figure. Which of the following statements is true?
$i. \triangle\text{ABC}\cong\triangle\text{ABD}$
$ii. \triangle\text{ABC}\cong\triangle\text{ADB}$
$iii. \triangle\text{ABC}\cong\triangle\text{BAD}$

Answer

In $\triangle\text{ABC}$ and $\triangle\text{BAD}$
we have, $\ce{AC = BD} ($given$) \ce{BC = AD} ($given$)$ and $\ce{AB = BA}($common$)$
Therefore, by $\ce{SSS}$ criterion of congruency,
$\triangle\text{ABC}\cong\triangle\text{BAD}$ There option $(iii)$ is true.

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