Question

Han was given that segment DC is parallel to segment AB and that angles A and C are right angles. Han wrote a

proof to show that triangle BCD is congruent to triangle DAB. Han's proof is wrong.
DC || AB
B
1) Line AB is parallel to DC and cut by transversal DB. So angles CDB and ABD are alterante interior angles and
must be congruent.
2) Side DB is congruent to side BD because they're the same segment.
3) Angle A is congruent to angle C because they're both right angles.
4) By the Angle-Side-Angle Triangle Congruence Theorem, triangle BCD is congruent to triangle DAB.
What is wrong with Han's proof?

Answer 1

The error in Han's proof is in the improper application of the Angle-Side-Angle (ASA) Triangle Congruence Theorem, as the side DB is not between the angles ABD and CDB, which is a requirement for ASA congruence.

Step-by-step explanation:

The student has asked what is wrong with Han's proof regarding the congruency of triangles BCD and DAB. The mistake in Han's proof lies in the conclusion made in step 4 based on the Angle-Side-Angle (ASA) Triangle Congruence Theorem. While it is true that angle C is congruent to angle A because they are both right angles and that segment BD is congruent to itself, Han erroneously concludes that just because angles ABD and CDB are congruent by alternate interior angles theorem, the triangles are congruent by ASA.

If we take a closer look at the criteria for ASA congruence, we see that we need two angles and a side between those angles to be congruent. In Han's proof, the congruent side BD is not between the two angles that were proven to be congruent but rather extends from one of these angles. Therefore, the ASA congruence criterion is not satisfied, and the proof is not valid as it stands.