Question

Select all statements that are true about the triangles.

Figure ABCD with diagonal BC, sides AB and DC are congruent, angle A is congruent to angle D and sides AC and DB are congruent.

Responses

A Triangles ABC
and DCB
are congruent by the Angle-Angle Triangle Congruence Theorem.Triangles cap A cap b cap c and cap d cap c cap b are congruent by the Angle-Angle Triangle Congruence Theorem.

B Triangles ABC
and BCD
are congruent by the Angle-Side-Angle Triangle Congruence Theorem.Triangles cap A cap b cap c and cap b cap c cap d are congruent by the Angle-Side-Angle Triangle Congruence Theorem.

C Triangles ABC
and BCD
are congruent by the Side-Side-Side Triangle Congruence Theorem.Triangles cap A cap b cap c and cap b cap c cap d are congruent by the Side-Side-Side Triangle Congruence Theorem.

D Triangles ABC
and DCB
are congruent by the Side-Angle-Side Triangle Congruence Theorem.Triangles cap A cap b cap c and cap d cap c cap b are congruent by the Side-Angle-Side Triangle Congruence Theorem.

E Triangles ABC
and DCB
are congruent by the Side-Side-Side Triangle Congruence Theorem.Triangles cap A cap b cap c and cap d cap c cap b are congruent by the Side-Side-Side Triangle Congruence Theorem.

F There is not enough information to determine if the triangles are congruent.There is not enough information to determine if the triangles are congruent.

Answer 1

The true statement about the congruence of triangles ABC and DCB is statement D, which applies the Side-Angle-Side (SAS) Triangle Congruence Theorem. This is because two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.

To determine whether triangles ABC and DCB are congruent, we can use several congruence theorems that involve comparing the triangles' sides and angles. Importantly, we need to identify which parts of the triangles are congruent based on the given information. As described, we have sides AB and DC that are congruent, angle A is congruent to angle D, and sides AC and DB are congruent.

Looking at the provided statements:

Statement A suggests Angle-Angle (AA) congruence, which is not a valid theorem for proving congruence of triangles.

Statement B is incorrect because it implies Angle-Side-Angle (ASA) congruence, but no information is given about the angle between sides AB and AC or sides DC and DB.

Statement C suggests Side-Side-Side (SSS) congruence, which doesn't apply since we only know that two sides from each triangle are congruent, not all three.

Statement D suggests Side-Angle-Side (SAS) congruence, which could be correct if the congruent angle is between the two pairs of congruent sides, which is the case here as angles A and D are between AB and AC, and DC and DB respectively.

Statement E suggests SSS congruence, again, which is incorrect due to the reason provided for statement C.

Statement F states that there is not enough information to determine congruency, which is incorrect because we do have enough information.

The correct statement is D, as it satisfies the SAS Triangle Congruence Theorem: two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, which means triangles ABC and DBD are congruent.