Given: angle BAC is congruent to angle ACD, segment BD bisects segment AC, and segment AC bisects angle BCD.
Prove: quadrilateral ABCD is a rhombus.
To prove that quadrilateral ABCD is a rhombus, we need to show that all four sides are congruent.
Given that angle BAC is congruent to angle ACD, this implies that triangle ABC is similar to triangle ACD because they share an angle and have an equal included angle. This means that angles ABC and CAD are congruent.
Since segment AC bisects angle BCD, this implies that angles ABC and ACD are congruent because they are opposite angles formed by intersecting lines.
Now, by the Angle Bisector Theorem, we know that segment BD bisects segment AC. This means that segments AB and CD are congruent because they are the two smaller halves of the larger segment AC. Additionally, segments AD and BC are congruent for the same reason.
So far, we have shown that the opposite sides AB and CD, as well as AD and BC, are congruent.
To prove that the remaining sides, AB and BC, are congruent, we will use the SAS (Side-Angle-Side) congruence theorem.
We have already established that angle ABC is congruent to angle ACD. Now, since we know that segments AB and CD are congruent, we have two pairs of congruent adjacent sides and the included angle between them. Therefore, triangle ABC must be congruent to triangle ACD by the SAS congruence theorem.
As a result, all four angles of quadrilateral ABCD are congruent, and all four sides are congruent. Therefore, quadrilateral ABCD is a rhombus.