Complete the proof by filling in the blanks. (If you would like to write it on paper, you may take a picture of your work and submit it)Given: ∠N≅∠S, line l bisects TR¯¯¯¯¯¯¯ at QProve: ΔNQT≅ΔSQR(4 points)

Answer 1

To complete the proof \( \Delta NQT \cong \Delta SQR \), we will utilize the information provided and the properties of congruent triangles. Here are the steps to fill in the blanks for the proof:

Given:

1. \( \angle N \cong \angle S \)
2. Line \( l \) bisects \( \overline{TR} \) at point \( Q \)

To Prove: \( \Delta NQT \cong \Delta SQR \)

Proof:

1. Given Information:

   • \( \angle N \cong \angle S \) (Statement given)
   • \( \overline{TQ} \cong \overline{RQ} \) (Since line \( l \) bisects \( \overline{TR} \) at \( Q \))

2. Angles:

   • By the definition of an angle bisector, we know that \( \angle NQT \cong \angle SQR \) (If line \( l \) is a bisector, then angles opposite the bisector are equal)

3. SAS Congruence:

   • Now we have:
     • \( \angle NQT \cong \angle SQR \) (from step 2)
     • \( \angle N \cong \angle S \) (from given)
     • \( \overline{TQ} \cong \overline{RQ} \) (from step 1)

4. Conclusion:

   • By the Side-Angle-Side (SAS) congruence theorem, we can conclude that:
     • \( \Delta NQT \cong \Delta SQR \)

Final Statement: Thus, \( \Delta NQT \cong \Delta SQR \) is proven.

This set of steps provides a complete logical flow from the given statements to the conclusion about the congruence of the two triangles.