Question

in \\( \triangle vwx \\), \\( \overline { vx } \\) is extended through point \\( x \\) to point \\( y \\), \\( m \angle w x y = ( 6 x + 11 ) ^ { \circ } \\), \\( m \angle xvw = ( 2 x + 14 ) ^ { \circ } \\), and \\( m \angle vwx = ( 2 x + 9 ) ^ { \circ } \\). find \\( m \angle w x y \\).

Explanation:

Step1: Recall the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In $\triangle VWX$, $\angle WXY$ is an exterior angle, and the two non - adjacent interior angles are $\angle XVW$ and $\angle VWX$. So we have the equation:
$m\angle WXY=m\angle XVW + m\angle VWX$

Step2: Substitute the given angle expressions into the equation

We know that $m\angle WXY=(6x + 11)^{\circ}$, $m\angle XVW=(2x + 14)^{\circ}$, and $m\angle VWX=(2x+9)^{\circ}$. Substituting these into the equation from Step 1, we get:
$6x + 11=(2x + 14)+(2x + 9)$

Step3: Simplify the right - hand side of the equation

Simplify $(2x + 14)+(2x + 9)$:
$2x+14 + 2x+9=(2x+2x)+(14 + 9)=4x+23$
So our equation becomes $6x + 11=4x+23$

Step4: Solve for $x$

Subtract $4x$ from both sides of the equation:
$6x-4x + 11=4x-4x+23$
$2x+11 = 23$
Subtract 11 from both sides:
$2x+11-11=23 - 11$
$2x=12$
Divide both sides by 2:
$x=\frac{12}{2}=6$

Step5: Find $m\angle WXY$

Substitute $x = 6$ into the expression for $m\angle WXY$, which is $(6x + 11)^{\circ}$:
$m\angle WXY=6\times6+11=36 + 11=47^{\circ}$

Answer:

The measure of $\angle WXY$ is $\boldsymbol{47^{\circ}}$.