Question

in the figure below, △abc ≅ △def
find the following
ab=
∠edf=

Explanation:

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

Since \(\triangle ABC\cong\triangle DEF\), then \(AB = DE\). So \(3x + 6=4x\).
Solve the equation \(3x + 6 = 4x\):
Subtract \(3x\) from both sides: \(6=4x - 3x\), so \(x = 6\).

Step2: Calculate the length of \(AB\)

Substitute \(x = 6\) into \(AB=3x + 6\).
\(AB=3\times6+6=18 + 6=24\) ft.

Step3: Use the property of congruent triangles (corresponding angles are equal)

Since \(\triangle ABC\cong\triangle DEF\), then \(\angle A=\angle D\).
We know \(x = 6\), so \(\angle A=6y+12\). Substitute \(x = 6\) into \(DE = 4x\), \(DE=24\) (but we use the angle - side relationship. First, since \(AB = 24\), \(DE = 24\), and from \(\angle A=\angle D\).
We know \(\angle A=6y + 12\), and \(\angle D=8y+2\).
Set \(6y + 12=8y+2\).
Subtract \(6y\) from both sides: \(12=2y+2\).
Subtract \(2\) from both sides: \(10 = 2y\), so \(y = 5\).
Then \(\angle EDF=\angle A\), substitute \(y = 5\) into \(\angle A=6y+12\).
\(\angle A=6\times5+12=30 + 12=42^{\circ}\).

Answer:

\(AB = 24\) ft, \(\angle EDF=42^{\circ}\)