In quadrilateral ABCD, the diagonals intersect at point T. Jasmine has used the Alternate Interior Angles Theorem to show that angle DAC is congruent to angle BCA and that angle ADB is congruent to angle CBD.

Which of the following can Jasmine use to prove that triangle ATD is congruent to triangle CTB?

AB ≅ DC AC ≅ DB DA ≅ BC AC ≅ AC

Jasmine can use **\( AC \cong AC \)** to prove that triangle \( \triangle ATD \) is congruent to triangle \( \triangle CTB \). This follows from the **Reflexive Property** of congruence, which states that any geometric figure is congruent to itself. In this case, segment \( AC \) is congruent to itself, and this can be used in conjunction with the angle congruences that Jasmine has already established.

Using **Angle-Side-Angle (ASA) congruence**, Jasmine can now conclude that the two triangles \( \triangle ATD \) and \( \triangle CTB \) are congruent because:
1. \( \angle DAC \cong \angle BCA \) (from the Alternate Interior Angles Theorem),
2. \( AC \cong AC \) (by the Reflexive Property),
3. \( \angle ADB \cong \angle CBD \) (from the Alternate Interior Angles Theorem).

### Do you want further details on this reasoning or have any questions?

Here are 5 related questions to expand the discussion:
1. What is the **Angle-Side-Angle (ASA)** congruence theorem?
2. How does the **Alternate Interior Angles Theorem** work in parallel lines?
3. How can Jasmine show that quadrilateral \( ABCD \) is a parallelogram?
4. What are other methods for proving triangle congruence besides ASA?
5. How does the **Reflexive Property** apply in various geometric proofs?

**Tip**: When proving triangle congruence, always look for congruent sides or angles that can be justified using properties like the Reflexive Property or congruence theorems.

In quadrilateral ABCD, the diagonals intersect at point T. Jasmine has used the Alternate Interior Angles Theorem to show that angle DAC is congruent to angle BCA and that angle ADB is congruent to angle CBD. Which of the following can Jasmine use to prove that triangle ATD is congruent to triangle CTB? AB ≅ DC AC ≅ DB DA ≅ BC AC ≅ AC

Learn how to prove triangle ATD is congruent to triangle CTB in quadrilateral ABCD using the Alternate Interior Angles Theorem, Reflexive Property, and ASA congruence. Explore the concepts of congruent angles, side congruence, and reflexive properties in this geometry problem suitable for Grades 8-10.

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