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

Han's proof incorrectly applies the Angle-Side-Angle Congruence Theorem because the side BD is not the included side between the congruent angles A and C, and angles CDB and ABD in the triangles DAB and BCD. Therefore correct option is 4

Step-by-step explanation:

The error in Han's proof rests in the assumption made in point 4, where the Angle-Side-Angle (ASA) Congruence Theorem is incorrectly applied. According to the ASA Congruence Theorem, two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.

In Han's proof, the side DB (also written as BD) is not the included side between the congruent angles A and C, and the congruent alternate interior angles CDB and ABD because it is not between angles A and B or C and D in triangles DAB and BCD, respectively. Hence, the proof should instead identify the matching sides and angles that fulfill the criteria for the ASA Congruence Theorem or any other congruence theorem such as Side-Angle-Side (SAS) or Side-Side-Side (SSS) if applicable.