Grade IX - Mathematics

Author:
Rahul Chandra Saha
M.Sc., Mathematics

Angles

Sum property of a triangle

Theorem: The sum of the angles of a triangle is $180^{\circ}$.

Proof:
Let $\mathrm{ABC}$ be a triangle. We have to show that $\angle A B C+\angle B A C+\angle A C B=$ $180^{\circ}$
FG parallel to BC is drawn. Then,
$\angle F A B=\angle A B C$
and
$\angle G A C=\angle A C B$
Now $\angle F A B+\angle B A C+\angle G A C=180^{\circ}$
Therefore,
\[
\angle A B C+\angle B A C+\angle A C B=180^{\circ} .(\mathrm{Q.E.D.})
\]
Question: Is there any relationship between the exterior angle $\angle \mathrm{ACE}$ and the interior angles of a triangle?

Theorem.
If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

Proof:

Let ABC be a triangle. We have to show that, $\angle B A C+\angle A B C=\angle A C E$
Look, $\angle A B C+\angle B A C+\angle A C B=180^{\circ}$
Also, $\angle A C B+\angle A C E=180^{\circ}$
Therefore,
\[
\angle A B C+\angle B A C+\angle A C B=\angle A C B+\angle A C E \cdot(\mathrm{Q} . \mathrm{E.D.})
\]