Question

if ∠rsu ≅ ∠tsu, ru = 4p - 65, and tu = 2p - 5, what is ru?

Explanation:

Step1: Use congruent - angle property

Since $\angle RSU\cong\angle TSU$ and $\angle R = \angle T=90^{\circ}$, and $SU$ is common, by the AAS (Angle - Angle - Side) congruence criterion, $\triangle RSU\cong\triangle TSU$. Then $RU = TU$.

Step2: Set up the equation

Set $4p - 65=2p - 5$.

Step3: Solve for $p$

Subtract $2p$ from both sides: $4p-2p - 65=2p-2p - 5$, which simplifies to $2p-65=-5$.
Add 65 to both sides: $2p-65 + 65=-5 + 65$, so $2p=60$.
Divide both sides by 2: $p = 30$.

Step4: Find $RU$

Substitute $p = 30$ into the expression for $RU$: $RU=4p - 65=4\times30-65$.
$RU = 120 - 65=55$.

Answer:

55