Question

write an indirect proof for the following conditional about the figure. if ∠efg and ∠hfj are vertical angles, then x≠3y. to prove the given conditional by contradiction, assume that ∠efg and ∠hfj are vertical angles and that x = 3y. because ∠efg and ∠hfj are vertical angles, they are congruent. this means their measures are equal, yielding the equation x + y=2x - y. (type an equation using x and y as the variables. do not include the degree symbol in your answer. do not simplify.) using the assumption to rewrite this equation in terms of y alone and then separately simplifying each side produces the equation 4y = 5y. it follows that y = 0, and so x = 0. (simplify your answers.) this means that m∠efg= and m∠hfj=. (type integers or decimals. do not include the degree symbol in your answers.)

Explanation:

Step1: Recall vertical - angle property

Since $\angle EFG$ and $\angle HFJ$ are vertical angles, $m\angle EFG=m\angle HFJ$, so $x + y=2x - y$.

Step2: Substitute $x = 3y$ into the equation

Substitute $x = 3y$ into $x + y=2x - y$. We get $(3y)+y=2(3y)-y$.
Simplify the left - hand side: $(3y)+y = 4y$.
Simplify the right - hand side: $2(3y)-y=6y - y=5y$. So $4y = 5y$.

Step3: Solve for $y$

Subtract $4y$ from both sides of $4y = 5y$, we get $0=y$.
Since $x = 3y$ and $y = 0$, then $x = 0$.

Step4: Find the measures of the angles

If $x = 0$ and $y = 0$, then $m\angle EFG=x + y=0+0 = 0$ and $m\angle HFJ=2x - y=2\times0-0 = 0$.

Answer:

$m\angle EFG = 0$, $m\angle HFJ = 0$