91Ó°ÊÓ

If the sum of measure of two angles is then they are said to be supplementary angles.

From the given figure it can be observed that,

$∠6$and $∠9$are supplementary angles, therefore,$m∠6+m∠9=180…\left(1\right)$

$∠7$and $∠10$are supplementary angles, therefore,$m∠7+m∠10=180…\left(2\right)$

$∠8$and are supplementary angles, therefore,$m∠8+m∠11=180…\left(3\right)$

Add equations (1),(2) and (3).

$$\begin{gathered} m∠6+m∠9+m∠7+m∠10+m∠8+m∠11=180+180+180 \\ m∠6+m∠7+m∠8+m∠9+m∠10+m∠11=540 \end{gathered}$$

The sum of measures of all interior angles of a triangle is 180.

From the given figure, it can be observed that $∠6,∠7$and $∠8$are interior angles of a triangle. Therefore,

$m∠6+m∠7+m∠8=180$

Substitute for $m∠6+m∠7+m∠8$into the equation obtained in step 4 to find the value of $m∠9+m∠10+m∠11$.

$$\begin{gathered} 180+m∠9+m∠10+m∠11=540 \\ m∠9+m∠10+m∠11=540−180 \\ =360 \end{gathered}$$

Therefore, $m∠9+m∠10+m∠11=360$.