Question

in the given geometric figure, triangle edc with angle at e being 70°, side ed is labeled 2x + 12, and segment cq is labeled 12x + 12. points e, c, q are colinear with an arrow from c to q.

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In triangle \( ECD \), the exterior angle \( \angle ECQ=12x + 12 \), and the two non - adjacent interior angles are \( \angle E = 70^{\circ}\) and \( \angle D=2x + 12\). So, \(12x+12=(2x + 12)+70\).

Step2: Solve the equation for \(x\)

First, simplify the right - hand side of the equation: \((2x + 12)+70=2x+82\).
The equation becomes \(12x+12 = 2x+82\).
Subtract \(2x\) from both sides: \(12x-2x+12=2x-2x + 82\), which simplifies to \(10x+12=82\).
Subtract \(12\) from both sides: \(10x+12 - 12=82 - 12\), so \(10x=70\).
Divide both sides by \(10\): \(x=\frac{70}{10}=7\).

Answer:

\(x = 7\)