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

Explanation:

Step1: Substitute x = 3y into the equation

Substitute \(x = 3y\) into \(x + y=2x - y\).
\(3y + y=2(3y)-y\)

Step2: Simplify both sides of the equation

Simplify the left - hand side: \(3y + y = 4y\).
Simplify the right - hand side: \(2(3y)-y=6y - y = 5y\).
So the equation becomes \(4y=5y\).

Step3: Find the contradiction

Subtract \(4y\) from both sides of \(4y = 5y\), we get \(0 = y\). Then \(x=3y = 0\). But if \(x = y=0\), the angles \(\angle EFG\) and \(\angle HFJ\) would be \(0^{\circ}\) which is a non - typical case for vertical angles in a general geometric sense. In a more general context, the equality \(4y = 5y\) is a contradiction since \(4y-5y=-y
eq0\) for non - zero values of \(y\).

Answer:

\(4y = 5y\)