Question

if ( rs = ru = 85 ) and ( mangle rtu = 56^{circ} ), what is ( mangle stu )?
( mangle stu=square^{circ} )

Explanation:

Step1: Apply the HL (Hypotenuse - Leg) Congruence Theorem

Since \(RS = RU = 85\), \(RT\) is the common hypotenuse (\(RT = RT\)), and \(\angle RST=\angle RUT = 90^{\circ}\), by the HL congruence theorem, \(\triangle RST\cong\triangle RUT\).

Step2: Use the property of congruent triangles

If \(\triangle RST\cong\triangle RUT\), then \(\angle STR=\angle UTR\). Let \(\angle STR = \angle UTR=x\).
We know that \(\angle RTU=\angle STR+\angle UTR\), and given \(\angle RTU = 56^{\circ}\). So \(2x = 56^{\circ}\).

Step3: Solve for \(x\)

Divide both sides of the equation \(2x = 56^{\circ}\) by \(2\). We get \(x=\frac{56^{\circ}}{2}=28^{\circ}\).

Answer:

\(28\)