Question

gle congruence
the proof that ( \triangle qptcong\triangle qrt ) is shown.
given: ( spcong sr )
line segment ( qt )
prove: ( \triangle qptcong\triangle qrt )

1. ( overline{qt}perpoverline{pr} )
2. ( \triangle ) and ( \triangle ) name the same line.
3. ( overline{qp}congoverline{qr} )
4. perpendicular bisector theorem
5. ( \triangle qptcong\triangle qrt )
6. hl theorem

reflexive property
definition of perpendicular bisector
definition of congruence
substitution property

Explanation:

Step1: Analyze the given information

We are given that \( \overline{QP}\cong\overline{QR}\) (from step 5) and \( \overline{QT}\perp\overline{PR}\) (from step 4). Also, \( \overline{QT}\) is a common side for \( \triangle QPT\) and \( \triangle QRT\).

Step2: Recall the HL (Hypotenuse - Leg) theorem

The HL theorem states that if the hypotenuse and a leg of a right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. In \( \triangle QPT\) and \( \triangle QRT\), \( \overline{QP}\) and \( \overline{QR}\) are the hypotenuses (\( \overline{QP}\cong\overline{QR}\)), \( \overline{QT}\) is the common leg, and \( \angle QTP=\angle QTR = 90^{\circ}\) (since \( \overline{QT}\perp\overline{PR}\)).

Answer:

\(\triangle QPT\cong\triangle QRT\) by the HL (Hypotenuse - Leg) theorem.