Given: SP ≅ SR
line segment
Prove: ΔQPT ≅ ΔQRT
Triangle P Q R is shown. Angle P Q R is cut by a perpendicular bisector to form midpoint T on side P R. Point S is on line Q S. Lines are drawn from points P and R to point S. Line segments P S and S R are congruent.
What is the missing reason in the proof?
Statements Reasons
1. SP ≅ SR 1. given
2. ST ⊥ PR 2. converse of the perpendicular bisector theorem
3. PT ≅ RT 3. ?
4. QT ⊥ PR 4. ST and QT name the same line.
5. QP ≅ QR 5. perpendicular bisector theorem
6. ΔQPT ≅ ΔQRT 6. HL theorem
definition of perpendicular bisector
definition of congruence
reflexive property
substitution property
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To fill in the missing reason for the statement 3, PT ≅ RT, we recognize that T is the midpoint of segment PR due to the definition of a perpendicular bisector.
Given that ST is a perpendicular bisector of PR, we can conclude that point T divides segment PR into two equal segments, PT and RT. Therefore, we can write:
3. PT ≅ RT
Reason: Definition of a midpoint.
The definition of a midpoint states that if T is the midpoint of segment PR, then the segments PT and RT are congruent.
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