Question

1.) write a congruence statement for the following figures: a. b. 2.) in the diagram below, defg ≅ spqr. a. find the value of x b. find the value of y.

Explanation:

Step1: Use the property of congruent figures

Since \(DEFG\cong SPQR\), their corresponding sides and angles are equal. For part a, the side \(DE = 8\) ft and the corresponding side \(QR=(2x - 4)\) ft. So, we set up the equation \(2x-4 = 8\).

Step2: Solve the equation for \(x\)

Add \(4\) to both sides of the equation \(2x-4 = 8\):
\(2x-4 + 4=8 + 4\)
\(2x=12\)
Divide both sides by \(2\):
\(x=\frac{12}{2}=6\)

Step3: Use the property of congruent figures for angles

The angle \(\angle D = 102^{\circ}\) and the corresponding angle \(\angle Q=(6y + x)^{\circ}\). Substitute \(x = 6\) into \((6y + x)\):
\(6y+6 = 102\)

Step4: Solve the equation for \(y\)

Subtract \(6\) from both sides of the equation \(6y+6 = 102\):
\(6y+6-6=102 - 6\)
\(6y=96\)
Divide both sides by \(6\):
\(y=\frac{96}{6}=16\)

Answer:

a. \(x = 6\)
b. \(y = 16\)