Question

the proof that $\triangle qpt\cong\triangle qrt$ is shown.
given: $sp\cong sr$
line segment $qt$
prove: $\triangle qpt\cong\triangle qrt$

1. $\overline{qt}\perp\overline{pr}$ 4. $s1$ and $q1$ name the same line.
2. $\overline{qp}\cong\overline{qr}$ 5. perpendicular bisector theorem
3. $\triangle qpt\cong\triangle qrt$ 6. hl theorem

reflexive property
definition of perpendicular bisector
definition of congruence
substitution property

Explanation:

Step1: Recall the HL (Hypotenuse - Leg) theorem

The HL theorem states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent.

Step2: Identify the right - triangles and their parts

In \(\triangle QPT\) and \(\triangle QRT\), \(\overline{QT}\perp\overline{PR}\), so \(\angle QTP=\angle QTR = 90^{\circ}\). We know from step 5 that \(\overline{QP}\cong\overline{QR}\) (hypotenuses) and \(\overline{QT}\) is a common leg (\(\overline{QT}\cong\overline{QT}\) by the reflexive property of congruence).

Answer:

reflexive property