Question
A diagonal of a parallelogram divides it into two congruent triangles.
✓
Answer
Proof : Let $\mathrm{ABCD}$ be a parallelogram and $\mathrm{AC}$ be a diagonal (see Fig. 8.2). Observe that the diagonal $\mathrm{AC}$ divides parallelogram $\mathrm{ABCD}$ into two triangles, namely, $\triangle \mathrm{ABC}$ and $\triangle \mathrm{CDA}$. We need to prove that these triangles are congruent.
In $\triangle \mathrm{ABC}$ and $\triangle \mathrm{CDA}$, note that $\mathrm{BC} \| \mathrm{AD}$ and $\mathrm{AC}$ is a transversal.
So, $\quad \angle \mathrm{BCA}=\angle \mathrm{DAC}$ (Pair of alternate angles)
Also, $\mathrm{AB} \| \mathrm{DC}$ and $\mathrm{AC}$ is a transversal.
So, $\quad \angle \mathrm{BAC}=\angle \mathrm{DCA}$ (Pair of alternate angles)
and $\mathrm{AC}=\mathrm{CA} \quad$ (Common)
So, $\triangle \mathrm{ABC} \cong \triangle \mathrm{CDA} \quad$ (ASA rule)
or, diagonal $\mathrm{AC}$ divides parallelogram $\mathrm{ABCD}$ into two congruent triangles $\mathrm{ABC}$ and $\mathrm{CDA}$.
Now, measure the opposite sides of parallelogram $A B C D$. What do you observe?
You will find that $\mathrm{AB}=\mathrm{DC}$ and $\mathrm{AD}=\mathrm{BC}$.
This is another property of a parallelogram stated below:
In $\triangle \mathrm{ABC}$ and $\triangle \mathrm{CDA}$, note that $\mathrm{BC} \| \mathrm{AD}$ and $\mathrm{AC}$ is a transversal.
So, $\quad \angle \mathrm{BCA}=\angle \mathrm{DAC}$ (Pair of alternate angles)
Also, $\mathrm{AB} \| \mathrm{DC}$ and $\mathrm{AC}$ is a transversal.
So, $\quad \angle \mathrm{BAC}=\angle \mathrm{DCA}$ (Pair of alternate angles)
and $\mathrm{AC}=\mathrm{CA} \quad$ (Common)
So, $\triangle \mathrm{ABC} \cong \triangle \mathrm{CDA} \quad$ (ASA rule)
or, diagonal $\mathrm{AC}$ divides parallelogram $\mathrm{ABCD}$ into two congruent triangles $\mathrm{ABC}$ and $\mathrm{CDA}$.
Now, measure the opposite sides of parallelogram $A B C D$. What do you observe?
You will find that $\mathrm{AB}=\mathrm{DC}$ and $\mathrm{AD}=\mathrm{BC}$.
This is another property of a parallelogram stated below:
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