Question

given: $qt = su$, $rt = tu$. prove: $qr = st$. statement reason 1 $qt = qr + rt$ segment addition property 2 $su = st + tu$ segment addition property 3 $qt = su$ given 4 $qr + rt = st + tu$ definition of congruent angles 5 $rt = tu$ definition of congruent angles 6 $qr + tu = st + tu$ addition and subtraction properties 7 $qr = st$ transitive property

Explanation:

Step1: Apply segment - addition property

$QT = QR+RT$ (Segment Addition Property) and $SU = ST + TU$ (Segment Addition Property)

Step2: Use given condition

Since $QT = SU$ (Given), we can substitute to get $QR + RT=ST + TU$

Step3: Substitute equal - length segments

Given $RT = TU$, substitute $RT$ with $TU$ in the equation $QR + RT=ST + TU$, we get $QR+TU = ST + TU$

Step4: Use subtraction property

Subtract $TU$ from both sides of the equation $QR+TU = ST + TU$. By the subtraction property of equality, we have $QR=ST$

Answer:

$QR = ST$ is proven as shown above.