## In a quadrilateral ABCD, the diagonals intersect at point T. Byron has used the Alternate Interior Angles Theorem to show that angle DAC is congruent to angle BCA and that angle BAC is congruent to DCA.Which of the following can Thomas use prove that side AD is equal to side BC?AC ≅ DBTB ≅ TDAC ≅ ACDB ≅ DB

### To prove that side AD is equal to side BC in quadrilateral ABCD, you can use the Side-Side-Side (SSS) Congruence Theorem

To prove that side AD is equal to side BC in quadrilateral ABCD, you can use the Side-Side-Side (SSS) Congruence Theorem. According to this theorem, if three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent.

In this case, we can consider triangles ADB and CBA. We need to show that all three sides of triangle ADB are congruent to the corresponding sides of triangle CBA.

1. AC ≅ DB (Given): This gives us one pair of congruent sides.

2. TB ≅ TD (Given): This gives us another pair of congruent sides.

3. Angle DAC ≅ Angle BCA (Given): This means that angle DAC and angle BCA are congruent, which gives us an angle-angle (AA) similarity between the two triangles.

4. Angle BAC ≅ Angle DCA (Given): This means that angle BAC and angle DCA are congruent, which confirms the AA similarity between the two triangles.

From the congruent angles, we can conclude that triangle ADB is similar to triangle CBA.

Now, since we have two pairs of congruent sides (AC ≅ DB and TB ≅ TD), we can use the SSS Congruence Theorem. Since both AC ≅ DB and TB ≅ TD, we can conclude that side AD is congruent to side BC.

Therefore, by using the given information and the SSS Congruence Theorem, we can prove that side AD is equal to side BC in quadrilateral ABCD.

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