Question

The sum of all the three angles in a triangle is

Answer 1

The sum of all the three angles in a triangle

A triangle is a polygon that has three sides and three vertices forming three angles

Consider a triangle $△\mathrm{ABC}$,

Draw a line segment $\mathrm{EF}$ Through point $A$ such that $EF\parallel BC$:

$\mathrm{EF}$

Now,

$\begin{array}{rcl}\because \mathrm{EF}& \parallel & \mathrm{BC}\\ \therefore \angle \mathrm{ABC}& =& \angle \mathrm{EAB}\left[\mathrm{Alternate}\mathrm{interior}\mathrm{angle}\right]\\ \mathrm{And}\angle \mathrm{BAC}& =& \angle \mathrm{FAC}\left[\mathrm{Alternate}\mathrm{interior}\mathrm{angle}\right]\end{array}$

Also, Since the $\angle \mathrm{EAB},\angle \mathrm{BAC}\mathrm{and}\angle \mathrm{FAC}$ are supplementary angles,

$$\begin{gathered} \therefore \angle \mathrm{EAB}+\angle \mathrm{BAC}+\angle \mathrm{FAC}=180° \\ \Rightarrow \angle \mathrm{ABC}+\angle \mathrm{CAB}+\angle \mathrm{ACB}=180°\left[\because \angle \mathrm{ABC}=\angle \mathrm{FAC}\mathrm{and}\angle \mathrm{BCA}=\angle \mathrm{FAC}\right] \end{gathered}$$

∴ The sum of all the three angles in a triangle is $180°$ .