Question

what values of v and w make \\( \triangle fgh \cong \triangle cab \\)?

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle FGH\cong\triangle CAB\), their corresponding sides are equal. So, \(FG = CA\) and \(GH=AB\).
For \(FG = CA\): \(13v - 14=v + 46\)

Step2: Solve the equation for \(v\)

\(13v-14 = v + 46\)
Subtract \(v\) from both sides: \(13v - v-14=v - v + 46\)
\(12v-14 = 46\)
Add \(14\) to both sides: \(12v-14 + 14=46 + 14\)
\(12v=60\)
Divide both sides by \(12\): \(v=\frac{60}{12}=5\)

Step3: Substitute \(v = 5\) into the equation for \(GH = AB\)

\(2w+v + 10=w-3v + 50\)
Substitute \(v = 5\): \(2w+5 + 10=w-3\times5 + 50\)
\(2w+15=w-15 + 50\)
\(2w+15=w + 35\)
Subtract \(w\) from both sides: \(2w - w+15=w - w + 35\)
\(w+15=35\)
Subtract \(15\) from both sides: \(w=35 - 15=20\)

Answer:

\(v = 5\), \(w = 20\)