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 DCA.

Which of the following can Heather use prove that segment DT is equal to segment TB?

AB ≅ DC
AC ≅ DB
DA ≅ BC
AC ≅ AC

Answer 1

To prove that segment DT is equal to segment TB in a quadrilateral's intersecting diagonals, Heather can use the reflexive property AC ≡ AC to establish the congruence of triangles ADC and ABC by SSS criteria.

Step-by-step explanation:

The student is working on a problem where they have to prove that in a quadrilateral ABCD with its diagonals intersecting at point T, segment DT is equal to segment TB. Heather can use the property AC ≡ AC to prove this fact by applying the Reflexive Property of Congruence, which states that any geometric figure is congruent to itself. Since AC is a diagonal that intersects with BD at point T, if we prove that the triangles ATC and BTC are congruent by SSS (Side-Side-Side) criteria, that would imply DT ≡ TB.

The criterion for SSS congruence is that three pairs of corresponding sides of triangles are equal, and since AC is equal to AC (it's the same line segment), Heather has already shown that AB ≡ DC and DB ≡ AC (as the alternate interior angles are congruent), we can conclude the triangles ADC and ABC are congruent by SSS and thus, DT ≡ TB.