Question

what values of a and b make \\( \triangle hij \cong \triangle edc \\)?

Explanation:

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

Since \(\triangle HIJ\cong\triangle EDC\), we have \(HI = ED\) and \(HJ=EC\).
So, \(b + 39=4b\) (because \(HI\) and \(ED\) are corresponding sides) and \(b + 2a+2=3b+a - 20\) (because \(HJ\) and \(EC\) are corresponding sides).

Step2: Solve the first - equation \(b + 39=4b\)

Subtract \(b\) from both sides: \(39=4b - b\), so \(3b=39\), then \(b = 13\).

Step3: Substitute \(b = 13\) into the second - equation \(b + 2a+2=3b+a - 20\)

Substitute \(b = 13\) into \(b + 2a+2=3b+a - 20\):
\(13+2a + 2=3\times13+a - 20\).
Simplify the left - hand side: \(15+2a\).
Simplify the right - hand side: \(39+a - 20=19+a\).
So, \(15+2a=19+a\).
Subtract \(a\) from both sides: \(15+a=19\).
Subtract 15 from both sides: \(a=4\).

Answer:

\(a = 4\), \(b = 13\)