Question

in \\( \triangle u v w, \overline{u w} \\) is extended through point \\( w \\) to point \\( x, \mathrm{m} \angle w u v=(2 x+15)^{\circ} \\), \\( \mathrm{m} \angle u v w=(2 x+15)^{\circ} \\), and \\( \mathrm{m} \angle v w x=(8 x-14)^{\circ} \\). find \\( \mathrm{m} \angle w u v \\).

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 $\triangle UVW$, $\angle VWX$ is an exterior angle, so $m\angle VWX=m\angle WUV + m\angle UVW$.

Given that $m\angle WUV=(2x + 15)^{\circ}$, $m\angle UVW=(2x + 15)^{\circ}$ and $m\angle VWX=(8x-14)^{\circ}$. Substituting these values into the exterior angle theorem formula, we get:

$$8x-14=(2x + 15)+(2x + 15)$$

Step2: Simplify the right - hand side of the equation

Simplify the right - hand side: $(2x + 15)+(2x + 15)=2x+2x + 15 + 15=4x+30$. So our equation becomes:

$$8x-14 = 4x+30$$

Step3: Solve for x

Subtract $4x$ from both sides: $8x-4x-14=4x - 4x+30$, which simplifies to $4x-14 = 30$.

Then add 14 to both sides: $4x-14 + 14=30 + 14$, so $4x=44$.

Divide both sides by 4: $x=\frac{44}{4}=11$.

Step4: Find $m\angle WUV$

We know that $m\angle WUV=(2x + 15)^{\circ}$. Substitute $x = 11$ into the expression:

$2x+15=2\times11 + 15=22 + 15=37$.

Answer:

$37^{\circ}$