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 resulting equation is \(4y = 5y\).

Step3: Analyze the resulting equation

Subtract \(4y\) from both sides of \(4y = 5y\), we get \(0 = y\). If \(y = 0\), then \(x=3y = 0\). But in general, vertical angles do not have measures of \(0\) (assuming non - degenerate cases). This is a contradiction to our assumption that \(x = 3y\) when \(\angle EFG\) and \(\angle HFJ\) are vertical angles.

Answer:

\(4y = 5y\)