Question

t is the midpoint of \\(\overline{rs}\\). complete the proof that \\(\angle s \cong \angle r\\).
image of a triangle with q, r, s vertices and t on rs, with qt perpendicular to rs

	statement	reason
1	t is the midpoint of \\(\overline{rs}\\)	given
2	\\(\overline{qt} \perp \overline{rs}\\)	given
3	\\(\angle qtr \cong \angle qts\\)	dropdown
4	\\(\overline{rt} \cong \overline{st}\\)	definition of midpoint
5	\\(\overline{qt} \cong \overline{qt}\\)	reflexive property of congruence
6	\\(\triangle qrt \cong \triangle qst\\)	sas
7	\\(\angle s \cong \angle r\\)	cpctc

Explanation:

Step1: Definition of perpendicular lines

Since \( \overline{QT}\perp\overline{RS}\), by the definition of perpendicular lines, \( \angle QTR = 90^{\circ}\) and \( \angle QTS=90^{\circ}\). Then, if two angles have the same measure (\(90^{\circ}\)), they are congruent. So, \( \angle QTR\cong\angle QTS\) (Right - angle congruence theorem: All right angles are congruent).

Answer:

Right - angle congruence theorem (All right angles are congruent)