Question

if ∠efh ≅ ∠gfh, eh = 2u - 6, and gh = 3u - 36, what is gh?

Explanation:

Step1: Use angle - side relationship

Since $\angle EFH\cong\angle GFH$, and $\angle E = \angle G=90^{\circ}$, and $FH$ is common to both $\triangle EFH$ and $\triangle GFH$, by the Angle - Angle - Side (AAS) congruence criterion, $\triangle EFH\cong\triangle GFH$. Then $EH = GH$.

Step2: Set up the equation

Set $2u - 6=3u - 36$.

Step3: Solve for $u$

Subtract $2u$ from both sides: $- 6=3u-2u - 36$, which simplifies to $-6 = u - 36$.
Add 36 to both sides: $u=-6 + 36=30$.

Step4: Find the value of $GH$

Substitute $u = 30$ into the expression for $GH$. $GH=3u - 36=3\times30-36$.
$GH = 90 - 36=54$.

Answer:

$54$