Question

what values of a and b make ( \triangle fgh cong \triangle kij )?

Explanation:

Step1: Use the property of congruent triangles (corresponding sides are equal)

Since \(\triangle FGH\cong\triangle KIJ\), then \(FG = KI\) and \(FH=KJ\).
For \(FG = KI\), we have the equation \(2a + 10=3a\).
Solve \(2a + 10=3a\):
Subtract \(2a\) from both sides: \(10=3a - 2a\), so \(a = 10\).

Step2: Substitute \(a = 10\) into the other equation

We know that \(FH=KJ\), so \(a + 8b-41=b + 6a\).
Substitute \(a = 10\) into \(a + 8b-41=b + 6a\):
\(10+8b-41=b + 6\times10\).
Simplify the equation: \(8b - 31=b + 60\).
Subtract \(b\) from both sides: \(8b - b-31=b - b + 60\), \(7b-31 = 60\).
Add \(31\) to both sides: \(7b=60 + 31\), \(7b=91\).
Divide both sides by \(7\): \(b=\frac{91}{7}=13\).

Answer:

\(a = 10\), \(b = 13\)