Question

if (mangle stv = mangle utv = 58^{circ}, sv=z + 64), and (uv = 9z), what is (sv)?

Explanation:

Step1: Identify congruent - triangles

Since $\angle STV=\angle UTV = 58^{\circ}$ and $\angle S=\angle U = 90^{\circ}$ (right - angles shown) and $TV$ is common to both $\triangle STV$ and $\triangle UTV$, by the Angle - Angle - Side (AAS) congruence criterion, $\triangle STV\cong\triangle UTV$.

Step2: Set up equation for equal sides

For congruent triangles, corresponding sides are equal. So $SV = UV$. Given $SV=z + 64$ and $UV = 9z$, we set up the equation $z + 64=9z$.

Step3: Solve for z

Subtract $z$ from both sides: $64=9z - z$, which simplifies to $64 = 8z$. Then divide both sides by 8: $z=\frac{64}{8}=8$.

Step4: Find SV

Substitute $z = 8$ into the expression for $SV$. Since $SV=z + 64$, then $SV=8 + 64=72$.

Answer:

72