What additional piece of information is needed to prove the triangles congruent by ASA?

2 scalene triangles JTB and SDP are given. Side JB and SP are equal in length and angle TBJ and SPD are equal is given.

A. ∠𝑇≅∠𝐷
B. ∠𝐽≅∠𝑆
C. 𝐽𝑇⎯⎯⎯⎯⎯⎯⎯⎯≅𝑆𝐷⎯⎯⎯⎯⎯⎯⎯⎯⎯
D. 𝑇𝐵⎯⎯⎯⎯⎯⎯⎯⎯≅𝐷𝑃⎯⎯⎯⎯⎯⎯⎯⎯⎯

1 answer

To prove the triangles JTB and SDP are congruent by the ASA (Angle-Side-Angle) criteria, we need two angles and the included side to be congruent.

From the information given:

  • We have \( \angle TBJ \cong \angle SPD \) (one pair of angles).
  • We have \( JB \cong SP \) (the included side between the angles).

To apply ASA, we need another angle to be congruent.

Among the options given:

A. \( \angle T \cong \angle D \) (This would give us another angle, which is needed.) B. \( \angle J \cong \angle S \) (This gives another angle, but it does not help in forming congruence within the requirement of the included side configuration.) C. \( JT \cong SD \) (This would provide a side, but we need another angle, not another side.) D. \( TB \cong DP \) (This also provides a side, which is not what we need.)

Thus, the correct answer is:

A. \( \angle T \cong \angle D \)

This gives the second angle needed to prove the triangles JTB and SDP are congruent by ASA.