Complete. $m\angle 9+m\angle 10+m\angle 11=?$

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,

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

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

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

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

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

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

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

$m\angle 6+m\angle 7+m\angle 8=180$

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

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

Therefore, $m\angle 9+m\angle 10+m\angle 11=360$.