Euclid I-11. (pg 9)

Emily Crum
Explanation: Draw line AB, with point C. Make any pt D on AC. Create pt E on CB such that CD=CE. Construct an equilateral triangle GDE with DE as its base. Connect GC. DC=CE as per the directions. DG=GE because tringle GDE is an equilateral triangle. GC = GC. Thus by SSS, triangle DGC is congruent to triangle EGC. This congruency means that angle DCG = angle ECG. Angles DCG and ECG are also supplementary because they form a striagnt line. Therefore, DCG + ECG = 180. Since these 2 angles are equal, let DCG=x=ECG. Then, 2x =180, x=90. Whenceforth, DCG and ECG are right angles and the line CG had been drawn at right angles to line AB with pt C on it.