The task involves geometrical constructions using a pair of compasses and a ruler. Here is the detailed explanation of each step:

Steps to Perform the Construction:

1. Draw Line Segment $AB$:

   • Draw a straight line segment $AB$ of length $7 \, \text{cm}$ using the ruler.
   • Place point $A$ at one end of the segment and $B$ at the other end.

2. Construct an Angle of $60^\circ$ at $A$:

   • Place the compass point on $A$ and draw an arc that intersects $AB$. Let this intersection be $P$.
   • Without changing the compass width, place the compass point on $P$ and draw another arc that intersects the first arc. Mark this intersection as $Q$.
   • Draw a straight line from $A$ through $Q$. This line forms a $60^\circ$ angle with $AB$.

3. Construct the Perpendicular Bisector of $AB$:

   • Place the compass point on $A$ and draw an arc above and below $AB$. The radius should be more than half the length of $AB$.
   • Without changing the compass width, place the compass point on $B$ and draw another set of arcs that intersect the previous ones. Mark the intersection points as $X$ and $Y$.
   • Draw a straight line through $X$ and $Y$. This is the perpendicular bisector of $AB$.
   • Mark the point of intersection of this bisector with $AB$ as $O$.

4. Construct the Angle Bisector of $\angle BAC$:

   • Place the compass point on $A$ and draw an arc that intersects both arms of $\angle BAC$. Mark the intersections as $R$ and $S$.
   • Place the compass point on $R$ and draw an arc within the angle. Repeat this from $S$, keeping the same compass width. Mark the intersection of these two arcs as $T$.
   • Draw a line from $A$ through $T$. This is the angle bisector of $\angle BAC$.

5. Construct a Circle with $O$ as the Center and $OB$ as Radius:

   • Place the compass point on $O$ and adjust the compass to the length of $OB$.
   • Draw a circle centered at $O$ with radius $OB$.

6. Join $A$ and $C$ and Measure $\angle ABC$:

   • Draw a straight line joining $A$ and $C$.
   • Measure $\angle ABC$ using a protractor.

Questions for Better Understanding:

1. How does constructing the perpendicular bisector help in dividing $AB$ into equal halves?
2. What is the significance of the angle bisector in this construction?
3. Why is it necessary to use a compass to construct a $60^\circ$ angle instead of using a protractor?
4. Can you explain why the intersection of the angle bisector and perpendicular bisector defines a unique circle's center?
5. How does the measurement of $\angle ABC$ relate to the constructed geometry?

Tip:

Always keep the compass steady and avoid changing its width while performing consecutive steps. This ensures accuracy in the construction.