Question

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

Short answer

$m\angle 9+m\angle 10+m\angle 11=\mathbf{360}$

Step-by-step solution

. Observe the given figure and highlight angle 9, angle 10 and angle 11. .

Step 2. Apply the definition of supplementary angles.

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

Step 3. Description of step.

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,$m\angle 8+m\angle 11=180\dots \left(3\right)$

Step 4. Description of step.

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

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

Step 5. Apply an angle sum theorem.

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

Step 6. Description of step.

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

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

Step 7. Description of step.

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