Question

∠1 and ∠2 are vertical angles. if (mangle1=(17x + 1)^{circ}) and (mangle2=(20x - 14)^{circ}), find (mangle2).

Explanation:

Step1: Recall vertical - angles property

Vertical angles are equal. So, $m\angle1 = m\angle2$.
Set up the equation: $17x + 1=12x - 14$.

Step2: Solve the equation for $x$

Subtract $12x$ from both sides: $17x-12x + 1=12x-12x - 14$, which simplifies to $5x+1=-14$.
Subtract 1 from both sides: $5x+1 - 1=-14 - 1$, getting $5x=-15$.
Divide both sides by 5: $x=\frac{-15}{5}=-3$.

Step3: Find $m\angle2$

Substitute $x = - 3$ into the expression for $m\angle2$: $m\angle2=12x-14$.
$m\angle2=12\times(-3)-14=-36 - 14=-50$ (This is wrong. Let's correct the setup. It should be $17x + 1=20x-14$).
Set up the correct equation: $17x + 1=20x-14$.
Subtract $17x$ from both sides: $17x-17x + 1=20x-17x-14$, which gives $1 = 3x-14$.
Add 14 to both sides: $1+14=3x-14 + 14$, so $15 = 3x$.
Divide both sides by 3: $x = 5$.
Substitute $x = 5$ into the expression for $m\angle2$: $m\angle2=20x-14$.
$m\angle2=20\times5-14=100 - 14=86^{\circ}$.

Answer:

$86^{\circ}$