Question

question 1
the figure below shows a quadrilateral abcd. sides ab and dc are equal and parallel.
a student wrote the following sentences to prove that quadrilateral abcd is a parallelogram.
side ab is parallel to side dc, so the alternate - interior angles, angle abd and angle cdb, are congruent. side ab is equal to side dc, and db is the side common to triangles abd and cdb. therefore, the triangles abd and cdb are congruent by sas postulate. by cpctc, angles dbc and bda are congruent and sides ad and bc are congruent. angle dbc and angle bda form a pair of vertical angles that are congruent. therefore ad is parallel and equal to bc. quadrilateral abcd is a parallelogram because its opposite sides are equal and parallel.
which statement best describes a flaw in the students proof?

1. triangles abd and bcd are congruent by the sss postulate instead of the sas postulate.
2. triangles abd and bcd are congruent by the aas postulate instead of the sas postulate.
3. angle dbc and angle bda form a pair of corresponding angles, not a pair of vertical angles.
4. angle dbc and angle bda form a pair of alternate interior angles that are congruent, not a pair of vertical angles.

Explanation:

The student claims that angles DBC and BDA are vertical angles, but they are actually alternate - interior angles. Vertical angles are formed by two intersecting lines, while alternate - interior angles are formed by a transversal intersecting two parallel lines.

Answer:

Angle DBC and angle BDA form a pair of alternate interior angles that are congruent, not a pair of vertical angles.