In a pair of adjacent angles (i) Vertex is always common (ii) One arm is…

CONCEPT: Two angles are called adjacent angles when they share a common vertex and exactly one common arm, with their interiors not overlapping. A linear pair is the special case of adjacent angles in which the two non-common (uncommon) arms lie along the same straight line, that is, they form opposite rays, making the two angles add up to 180 degrees.

APPLICATION:

1. Statement (i), vertex is always common: true, because sharing a vertex is part of the very definition of adjacent angles.

2. Statement (ii), one arm is always common: true, because sharing exactly one arm is likewise part of that definition.

3. Statement (iii), uncommon arms are always opposite rays: false in general. The opposite-rays condition is what turns an ordinary adjacent pair into the special case of a linear pair; it is an additional requirement, not a consequence of adjacency itself. Two adjacent angles can have their uncommon arms pointing in many directions, not necessarily opposite, as long as the vertex and one arm are shared and the interiors do not overlap.

CROSS-CHECK: Consider two adjacent angles of 30 degrees and 50 degrees sharing a vertex O and a common arm OB, with their other arms OA and OC making an angle of 80 degrees between them, not 180 degrees. Here (i) and (ii) clearly hold, and OA, OC are not opposite rays, so (iii) fails, confirming that only statement (iii) can fail while (i) and (ii) always hold.

Since only statement (iii) is false, the option identifying (iii) is false is correct.