Can two lines share the same vertex and not be congruent?

Answer 1

Yes,

Congurent lines are lines that have similar lengths, in geometry they are lines or shapes that can be flipped, rotated, or reflected onto each other. In analytic geometry two lines are congruent if they share any TWO poins. A vertex is ONE point. If you rotate the points outside of their plane through the Z axis then they are not longer congruent with any lines in the XY plane. But the lines can also be flipped and rotated in any fashion to make it not line up with something else.

The only way two lines with the same vertex can be congruent is if they share the same vector length. Or another words you can draw a sphere, centered on the common vertex, and connect both lines to the edge of the sphere. If one line is longer or shorter then it is no longer congruent.

In the standard 2D world a congruent line with the same vertex would touch the edge of a circle,around that vertex, of the same radius.

Vectors are used to calculate, among other things, how forces react to each other and change each other. For example if you provide a force of 1 lb/sec to an object and then apply a same force of 1 lb/sec at at 45 degee angle to the first force you will have a combined vector along the 45 degree angle. The object would be the vertex of the two forces. In the case of 1 lb/sec for each force they would be congruent, but not if one of the forces was increased or decreased.

Answer 2

If AB and CD are two straight lines intersecting at K, then the angles AKC and BKD are vertically opposite and equal to each other. The same is true of AKD and BKC.

However, if AKB is not a straight line or CKD is not a straight line, then the shared vertex K does not imply the angles are equal.

Answer 3

Yes, they can, if they are not the same length.