Question

Write a paragraph proof.

Given: $▱ABCD;▱BEDF$

Prove: AECF is a $▱$.

(Hint: A short proof is possible if certain auxiliary segments ate drawn.)

Short answer

Two pairs of opposite sides $\overrightarrow{AE}\cong \overrightarrow{FC}$and $\overrightarrow{AF}\cong \overrightarrow{CE}$in AECF are parallel therefore, AECF is a parallelogram.

Step-by-step solution

Step 1. Relation between triangle DAE and triangle BCF

In triangles$\Delta DAE$ and $\Delta BCF$.

$\overrightarrow{AD}\cong \overrightarrow{BC}\therefore$(ABCD is a rhombus)

$\angle ADC\cong \angle ABC\therefore$(ABCD is a rhombus)

$\angle ADE\cong \angle CBF\therefore$($\angle EDC\&\angle ABF$ are the equal angle of parallelogram made of parallel sides two parallelograms)

ABCD is a parallelogram.

Similarly,$\angle ADC\cong \angle ABC$ ( ABCD is a rhombus),

$\angle EDC\&\angle ABF$are equal angles of parallelogram made of parallel sides.

$\angle ADE\cong \angle CBF$

$\overrightarrow{DE}\cong \overrightarrow{BF}$$\left(\therefore BEDF\text{\hspace{0.17em}isa\hspace{0.17em}parallelogram}.\right)$

Therefore, by SAS postulate $\Delta DAE\cong \Delta BCF$.

By CPCT, $\overrightarrow{AE}\cong \overrightarrow{FC}$.

Step 2. Relation between triangle AFB and triangle CDE

In triangles $\Delta AFB\&\Delta CDE$.

$\overrightarrow{DE}\cong \overrightarrow{BF}\therefore \left(BEDF\text{\hspace{0.17em}isaparallelogram}\right)$

Since,$\angle EDC\&\angle ABF$ are equal angles of parallelogram made of parallel sides.

$\angle ADE\cong \angle CBF$

$\overrightarrow{DC}\cong \overrightarrow{BA}\therefore \left(BEDF\text{\hspace{0.17em}isaparallleogram}\right)$

Therefore, by SAS postulate $\Delta AFB\cong \Delta CED$.

By CPCT, $\overrightarrow{AF}\cong \overrightarrow{CE}$.

Step 3. State the conclusion

Since two pairs of opposite sides $\overrightarrow{AE}\cong \overrightarrow{FC}$and$\overrightarrow{AF}\cong \overrightarrow{CE}$ in AECF are parallel, therefore, AECF is a parallelogram.