Question

(02.06 mc) in quadrilateral abcd, the diagonals intersect at point t. thomas has used the alternate interior angles theorem to show that angle adb is congruent to angle dbc and that angle dba is congruent to angle bdc. which of the following can thomas use to prove that side ab is equal to side dc? (1 point) ac ≅ ac ac ≅ db ta ≅ tc db ≅ db

Answer 1

Thomas can use the statement "db ≅ db" to prove that side AB is equal to side DC.

Step-by-step explanation:

In quadrilateral ABCD, the diagonals intersect at point T, and Thomas has used the Alternate Interior Angles Theorem to establish angle ADB congruent to angle DBC and angle DBA congruent to angle BDC. Given these angle relationships, Thomas aims to show that side AB is equal to side DC. To achieve this, he needs a congruent side to AB. Among the given options, the statement "db ≅ db" indicates that side DB is congruent to itself.

According to the Reflexive Property of Congruence, any geometric figure is congruent to itself. Therefore, Thomas can use the fact that "db ≅ db" to apply the Side-Angle-Side (SAS) congruence criterion. By SAS, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. In this case, triangle ADB is congruent to triangle CDB. Consequently, side AB is equal to side DC due to the corresponding congruent parts of congruent triangles.

Understanding congruence criteria and properties allows students like Thomas to apply logical reasoning and geometric principles to prove various relationships within geometric figures, leading to a deeper comprehension of the underlying concepts in geometry.

Answer 2

the correct choice that Thomas can use to prove that side AB is equal to side DC is:

db ≅ db

Therefore, option D is correct

To prove that side AB is equal to side DC in quadrilateral ABCD where the diagonals intersect at point T, Thomas needs to use properties that show a pair of corresponding sides are congruent.

Given that angle ADB is congruent to angle DBC and angle DBA is congruent to angle BDC by the Alternate Interior Angles Theorem, Thomas can prove that triangle ADB is congruent to triangle DBC using the Angle-Side-Angle (ASA) postulate if he can show that the included side, DB, is congruent in both triangles. However, since DB is a common side to both triangles ADB and DBC, it is already congruent to itself.

Therefore, the fact that DB is congruent to itself (DB ≅ DB) can be used in conjunction with the congruent angles to show that triangle ADB is congruent to triangle DBC by the ASA postulate. Once the two triangles are shown to be congruent, it then follows that the corresponding sides AB and DC are congruent (AB ≅ DC) due to corresponding parts of congruent triangles being congruent (CPCTC).

So, the correct choice that Thomas can use to prove that side AB is equal to side DC is:

db ≅ db