Question

a triangle dfe with a right angle (90°) at f, angles at d labeled (3x + 2)° and at e labeled (2x + 18)°, and a line with point g passing through f, forming a straight line with one side of the triangle.

Explanation:

Step1: Recall triangle angle sum property

In a triangle, the sum of interior angles is \(180^\circ\). Also, \(\angle DFE\) is supplementary to the \(90^\circ\) angle, so \(\angle DFE = 90^\circ\) (since they form a linear pair? Wait, no, actually, the angle at \(F\) inside the triangle: since the line \(GF\) is a straight line, the angle adjacent to \(90^\circ\) (inside the triangle) is also \(90^\circ\)? Wait, no, looking at the diagram, the triangle \(DFE\) has angles at \(D\): \((3x + 2)^\circ\), at \(E\): \((2x + 18)^\circ\), and at \(F\): \(90^\circ\) (because the angle at \(F\) between \(DF\) and \(EF\) is such that the adjacent angle is \(90^\circ\), so they are complementary? Wait, no, linear pair: if one angle is \(90^\circ\), the other (inside the triangle) is also \(90^\circ\)? Wait, no, the sum of angles in a triangle is \(180^\circ\), so:

\((3x + 2) + (2x + 18) + 90 = 180\)

Step2: Simplify the equation

Combine like terms:

\(3x + 2 + 2x + 18 + 90 = 180\)

\(5x + 110 = 180\)

Step3: Solve for \(x\)

Subtract 110 from both sides:

\(5x = 180 - 110\)

\(5x = 70\)

Divide both sides by 5:

\(x = \frac{70}{5}\)

\(x = 14\)

Answer:

\(x = 14\)