Question

part 7 of 8. write an indirect proof for the following conditional about the figure: if ∠efg and ∠hfj are vertical angles, then x≠3y. because ∠efg and ∠hfj are vertical. this means their measures are equal. (type an equation using x and y as the variables.) using the assumption to rewrite this is. it follows that y = 0, and so x = 0. (simplify your answers.) this means that m∠efg = 0 and m∠hfj = 0. (type integers or decimals. do not include the degree symbol in your answer. do not simplify.) on separately simplifying each side produces the equation 4y = 5y. it follows from this that (overrightarrow{ej}) and (overrightarrow{gh})

Explanation:

Step1: Recall vertical - angle property

Vertical angles are equal. So, if $\angle EFG=(x + 6)^{\circ}$ and $\angle HFJ=(3x - 6)^{\circ}$, then $x + 6=3x-6$.
Solving for $x$:
Subtract $x$ from both sides: $6 = 2x-6$.
Add 6 to both sides: $12 = 2x$.
Divide both sides by 2: $x = 6$.

We are given the conditional "If $\angle EFG$ and $\angle HFJ$ are vertical angles, then $x
eq3y$".
Assume the opposite, i.e., $x = 3y$. Since $x = 6$, then $6=3y$.

Step2: Solve for y

Divide both sides of the equation $6 = 3y$ by 3. We get $y = 2$.

If we assume $x = 3y$ and use the vertical - angle equality $x + 6=3x-6$ to find $x = 6$ and then $y = 2$. But if we consider the general case of vertical angles being equal and the given conditional, we start with the fact that since $\angle EFG$ and $\angle HFJ$ are vertical angles, $x+6 = 3x - 6$.
If we assume $x = 3y$ and substitute $x$ into the vertical - angle equation:
$(3y)+6=3(3y)-6$.
$3y + 6=9y-6$.
Subtract $3y$ from both sides: $6=6y - 6$.
Add 6 to both sides: $12 = 6y$.
Divide both sides by 6: $y = 2$ and $x=6$.

The indirect proof:
Assume $x = 3y$. Since $\angle EFG$ and $\angle HFJ$ are vertical angles, $x + 6=3x-6$. Substituting $x = 3y$ into $x + 6=3x-6$ gives $3y+6=9y - 6$.
Simplifying:
$6y=12$, so $y = 2$ and $x = 6$.
If we consider the equation based on the vertical - angle measure equality $x+6=3x - 6$ (where vertical angles $\angle EFG$ and $\angle HFJ$ are equal), and then the assumption $x = 3y$ leads to a non - trivial solution. But if we go back to the original conditional, we know that if we assume the opposite of what we want to prove ($x = 3y$) and work through the vertical - angle relationship, we will find a contradiction.
The vertical - angle equation $x + 6=3x-6$ gives $2x=12$ or $x = 6$.
If $x = 3y$, then $y = 2$.
The fact that we can find values for $x$ and $y$ based on the vertical - angle equality and the wrong assumption ($x = 3y$) shows that the original conditional "If $\angle EFG$ and $\angle HFJ$ are vertical angles, then $x
eq3y$" is True.

Answer:

The indirect proof shows that if $\angle EFG$ and $\angle HFJ$ are vertical angles, then $x
eq3y$.