1. **Problem Statement:** Given triangles $\triangle ABC$ and $\triangle DEF$ with $AB = DE$, $AB \parallel DE$, $BC = EF$, and $BC \parallel EF$, and vertices $A, B, C$ joined to $D, E, F$ respectively, prove the following: 2. **(i) Show quadrilateral $ABED$ is a parallelogram:** - Since $AB = DE$ and $AB \parallel DE$, and $BE$ and $AD$ are the other two sides connecting these points, by definition, a quadrilateral with one pair of opposite sides equal and parallel is a parallelogram. - Therefore, $ABED$ is a parallelogram. 3. **(ii) Show quadrilateral $BEFC$ is a parallelogram:** - Given $BC = EF$ and $BC \parallel EF$, and $BE$ and $CF$ are the other sides, similarly, $BEFC$ has one pair of opposite sides equal and parallel. - Hence, $BEFC$ is a parallelogram. 4. **(iii) Show $AD \parallel CF$ and $AD = CF$:** - In parallelogram $ABED$, opposite sides $AD$ and $BE$ are parallel and equal. - In parallelogram $BEFC$, opposite sides $BE$ and $CF$ are parallel and equal. - Since $AD$ is parallel and equal to $BE$, and $BE$ is parallel and equal to $CF$, by transitivity, $AD \parallel CF$ and $AD = CF$. 5. **(iv) Show quadrilateral $ACFD$ is a parallelogram:** - From (iii), $AD \parallel CF$ and $AD = CF$. - Also, $AC$ and $DF$ are the other two sides. - If one pair of opposite sides in a quadrilateral are equal and parallel, the quadrilateral is a parallelogram. - Therefore, $ACFD$ is a parallelogram. 6. **(v) Show $AC = DF$:** - Since $ACFD$ is a parallelogram, opposite sides $AC$ and $DF$ are equal. - Hence, $AC = DF$. 7. **(vi) Show $\triangle ABC \cong \triangle DEF$:** - We have $AB = DE$, $BC = EF$, and $AC = DF$ from above. - By the Side-Side-Side (SSS) congruence criterion, $\triangle ABC \cong \triangle DEF$. **Final answers:** (i) $ABED$ is a parallelogram. (ii) $BEFC$ is a parallelogram. (iii) $AD \parallel CF$ and $AD = CF$. (iv) $ACFD$ is a parallelogram. (v) $AC = DF$. (vi) $\triangle ABC \cong \triangle DEF$.