Question

solve for x.

Explanation:

Step1: Assume $\angle BDC$ and $\angle BDA$ are supplementary

$\angle BDA=180-(9x + 2)$

Step2: Use angle - sum property of a triangle

In $\triangle BDC$, $\angle B+\angle BDC+\angle C = 180^{\circ}$. But we assume $\angle BDA$ and $\angle BDC$ are related as above. If we assume $\angle BDA$ and $\angle BDC$ are linear - pair and use the fact that in $\triangle BDC$, we can also consider the exterior - angle property. The exterior angle $\angle BDA$ of $\triangle BDC$ is equal to the sum of the two non - adjacent interior angles $\angle B$ and $\angle C$. So $9x + 2=75+(2x + 11)$.

Step3: Solve the equation

$9x+2=75 + 2x+11$
$9x-2x=75 + 11-2$
$7x=84$
$x = 12$

Let's assume we use the exterior - angle property correctly.
If we assume the figure has some parallel - line or other geometric relationship that is not clearly marked, and we just consider the equation based on the exterior - angle of the triangle formed:
$9x+2=75+(2x + 11)$
$9x+2=2x+86$
$9x-2x=86 - 2$
$7x=84$
$x = 12$

If we made a wrong assumption above and assume that the two angles $9x + 2$ and $2x+11$ are related in a different way. Let's assume they are equal (if there is some congruence or parallel - line property not shown).
$9x+2=2x + 11$
$9x-2x=11 - 2$
$7x=9$
$x=\frac{9}{7}$

Let's assume the correct relationship is based on the exterior - angle of the triangle.
$9x+2=75+(2x + 11)$
$9x+2=2x + 86$
$9x-2x=86 - 2$
$7x=84$
$x = 12$

Let's re - check our work. Substitute $x = 12$ into the expressions for the angles.
The angle $9x+2=9\times12 + 2=108 + 2=110$
The angle $2x+11=2\times12+11=24 + 11=35$
$75+35 = 110$

So the value of $x$ is $12$.

Step1: Apply exterior - angle property

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So $9x + 2=75+(2x + 11)$.

Step2: Simplify the equation

$9x+2=2x+86$.

Step3: Solve for $x$

Subtract $2x$ from both sides: $9x-2x+2=2x-2x + 86$, which gives $7x+2=86$. Then subtract 2 from both sides: $7x+2 - 2=86 - 2$, so $7x=84$. Divide both sides by 7: $x=\frac{84}{7}=12$.

Answer:

$x = 1$