Question

1. ( mangle2 = 4x - 2 )

Explanation:

Step1: Determine the type of triangles

The two triangles are congruent (by SSS - Side - Side - Side congruence as two sides are marked equal and the common side is equal). So, the base - angles of the left - hand triangle (where one angle is \(68^{\circ}\)) and the angle relationships in the right - hand part (where \(\angle2\) is involved) can be used. The triangle with the \(68^{\circ}\) angle is an isosceles triangle. The sum of angles in a right - angled isosceles triangle (formed by the congruent triangles) gives: \(2m\angle2+68^{\circ}=90^{\circ}\) (since the right - angle is \(90^{\circ}\)).

Step2: Solve for \(x\)

We know that \(m\angle2 = 4x−2\). From \(2m\angle2+68^{\circ}=90^{\circ}\), first solve for \(m\angle2\):

$$
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$$

Then set \(4x−2=11\). Add \(2\) to both sides: \(4x=11 + 2=13\). Divide both sides by \(4\): \(x=\frac{13}{4}=3.25\)

Answer:

\(x = 3.25\)