91Ó°ÊÓ

The given diagram is:

The theorem 3-12 states that the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

It is being given that:

$\begin{array}{c}m∠1+m∠2+m∠3=180\left(1\right)\\ m∠3+m∠4=180\left(2\right)\end{array}$

The theorem 3-12 is to be proved.

From the diagram it can be noticed that the exterior angle is $∠4$and the two remote interior angles are $∠1$and $∠2$.

Therefore, it is to be proved that:$m∠1+m∠2=m∠4$

From the equation (2), it can be obtained that:

$$\begin{gathered} m∠1+m∠2+m∠3=180\left(1\right) \\ m∠3+m∠4=180\left(2\right) \end{gathered}$$

Now, substitute the value of $m∠3$from the equation (3) in the equation (1).

Therefore, it is obtained that:

$$\begin{gathered} m∠1+m∠2+m∠3=180 \\ m∠1+m∠2+180−m∠4=180 \\ m∠1+m∠2=180−180+m∠4 \\ m∠1+m∠2=m∠4 \end{gathered}$$

Hence proved.