Question

12 what is the measure of angle c? a) 37 b) 106 c) 74 d) 180

Explanation:

Step1: Use congruent triangle properties

Since \(DE = DC\) and \(EC\) is common side, \(\triangle DEC\) is isosceles. But wait, no, actually, in the figure (assuming it's a congruent triangle setup, maybe \( \triangle DCE\cong\triangle BAE\) - but wait, no, looking at the angle \(\angle D = 106^{\circ}\). Wait, no, wait, in a triangle, the sum of angles is \(180^{\circ}\). If we assume \(\triangle DCE\) where \(DE = DC\) (marked as equal), then \(\angle E=\angle C\). Let \(\angle C = x\), \(\angle E=x\). Given \(\angle D = 106^{\circ}\).

Step2: Apply angle - sum formula

By the angle - sum property of a triangle (\(\angle D+\angle C+\angle E = 180^{\circ}\)), substitute \(\angle E=\angle C\). So \(106^{\circ}+x + x=180^{\circ}\), which simplifies to \(106^{\circ}+2x=180^{\circ}\). Then \(2x=180^{\circ}- 106^{\circ}=74^{\circ}\), and \(x = 37^{\circ}\).

Answer:

\(37^{\circ}\) (option a)