Question

in the diagram, \\( \triangle abc \cong \triangle def \\).

1. find the value of \\( x \\).
2. find the value of \\( y \\).

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle ABC\cong\triangle DEF\), corresponding angles and sides are equal.
For the sides: \(AC = DF\), \(AB=DE\), \(BC = EF\). For the angles: \(\angle A=\angle D\), \(\angle B=\angle E\), \(\angle C=\angle F\)

Step2: Find the value of \(x\)

We know that \(BC = EF\). Given \(BC = 90.6\) and \(EF=2x + y\). Also, \(\angle C=\angle F = 40^{\circ}\), \(\angle A=\angle D=55^{\circ}\), \(\angle B=\angle E = 75^{\circ}\)
Since \(BC = EF\), we have \(2x + y=90.6\)

Step3: Find the value of \(y\)

We know that \(\angle D=(2y - 5)^{\circ}\) and \(\angle A = 55^{\circ}\). Since \(\angle A=\angle D\) (because \(\triangle ABC\cong\triangle DEF\)), we set up the equation \(2y-5 = 55\)

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Step4: Substitute \(y = 30\) into the equation for \(x\)

Substitute \(y = 30\) into \(2x + y=90.6\)

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Answer:

1. \(x = 30.3\)
2. \(y = 30\)