Which construction could be used to construct an isosceles triangle ABC given line segment AB?
A. Mark a third point C not on segment AB. Draw segments AC and BC.
B. Label a point Con segment AB and construct a line perpendicular to AB
through point C. Draw segments AC and BC.
C. Construct the perpendicular bisector of segment AB. Mark the intersection of
this line and AB and label it C. Draw segments AC and BC.
D. Construct the perpendicular bisector of segment AB. Mark any point C on the
perpendicular bisector except where it intersects AB. Draw segments AC and
ВС.

20 points

An isosceles triangle is one which has two equal sides and angles. Thus for the construction question given, the answer is Option D.

An isosceles triangle has two equal sides and angles.  Thus the required steps to construct an isosceles triangle as given in the question are:

i. Draw the given line AB.

ii. With any radius (greater than the midpoint of AB) and point A, draw an arc above  AB.

iii. Using point B now and the same radius, draw arcs to intersect the previous arc.

iv. Mark any point C on the  perpendicular bisector except where it intersects AB. Draw segments AC and  ВС.

This will produce a triangle with equal base angles and two equal sides. Thus an isosceles triangle ABC.

Check: link

In the given two column proof two sides and included angles are equal.

Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles .In the figure sides BD≅BD , AB≅ BC and <ABD ≅,BDD Therefore triangle ABD and triangle CBD are congruent by SAS property of congruence.