Question

In a quadrilateral ABCD, the diagonals intersect at point T. Heather has used the alternate interior angles theorem to show that angle ABD is congruent to angle CDB and that angle BAC is congruent to angle DCA. Which of the following can Heather use to prove that segment DT is equal to segment TB?

1) AB ≅ DC
2) AC ≅ DB
3) DA ≅ BC
4) TA ≅ TC

Answer 1

To prove that segment DT is equal to segment TB, Heather should use the fact that segments (option 4) TA ≅ TC, supporting the congruence of triangles ATD and CTB by ASA postulate, which makes sides DT and TB congruent.

Step-by-step explanation:

The question is asking which piece of information can be used to prove that segment DT is equal to segment TB in a quadrilateral ABCD with diagonals intersecting at point T. It is given that angle ABD is congruent to angle CDB and angle BAC is congruent to angle DCA, using the alternate interior angles theorem. To prove that DT is equal to TB, Heather can use the fact that TA is congruent to TC (Option 4).

If the triangles ATD and CTB share a side (AC) and have two pairs of equal angles (the angles given by the alternate interior angles theorem), they would be congruent by ASA postulate (Angle-Side-Angle). This would therefore mean that the sides DT and TB which are corresponding parts of congruent triangles, are also congruent, i.e., equal in length.

Answer 2

3) DA ≅ BC. Heather can use the congruent triangles created by the alternate interior angles theorem to prove that segment DT is equal to segment TB. The option she can use is 3) DA ≅ BC.

Step-by-step explanation:

To prove that segment DT is equal to segment TB, Heather can use the congruent triangles created by the alternate interior angles theorem. If angle ABD is congruent to angle CDB, and angle BAC is congruent to angle DCA, then triangles ABD and DBC are congruent. This congruence implies that their corresponding sides are also congruent. Since AB is congruent to DC, and DB is congruent to DA, it follows that segment DT is equal to segment TB. Therefore, the option Heather can use to prove that segment DT is equal to segment TB is 3) DA ≅ BC.