write a two column proof for this
Given:QR||TU , S is the midpoint of QT
Prove: /\ QSR=~/\TSU
/\ IS the triangle sign
=~ is the congruent symbol
Statement │ Reason
--------------------------------------------------------
1. QR || TU │ Given
2. QS = SR │ S is the midpoint of QT
3. QS || SR │ If a line is divided into two equal parts, the segments are parallel
4. QS || TS │ Transitive Property of Parallel Lines
5. QS = TS │ Corresponding sides of parallel lines are congruent
6. /\ QSR = /\ TSU │ SAS Postulate, since QS = TS and /\ QSR and /\ TSU share side SR.
Therefore, /\ QSR is congruent to /\ TSU, as proven.
Statements | Reasons
1. QR || TU | Given
2. QS = SR | S is the midpoint of QT
3. QS || TR | Opposite sides of a parallelogram are parallel
4. /\ QSR ≅ /\ TSQ | SAS (Side-Angle-Side) congruence
5. /\ QSR ≅ /\ TSU | CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
To write a two-column proof for this problem, we will list the statements and reasons in two separate columns. The statements will consist of the given information and the logical steps we take to prove the desired congruency. Here's the proof:
| Statements | Reasons |
|--------------------------|------------------------------------------------|
| 1. QR || TU | Given |
| 2. S is the midpoint of QT | Given |
| 3. QS = SR | Definition of the midpoint |
| 4. ∠QSR = ∠TSU | Corresponding angles in parallel lines |
| 5. QS = TS | Substitution (from statement 3 and 2) |
| 6. ∠QSR = ∠TSU | Substitution (from statement 4) |
| 7. ΔQSR ≅ ΔTSU | Angle-Side-Angle (ASA) congruence |
| 8. ∠QSR ≅ ∠TSU | Corresponding parts of congruent triangles |
| Δ QSR ≅ Δ TSU | Angle-Side-Angle (ASA) congruence | (Conclusion)
In the proof, we use the definitions of a midpoint and corresponding angles in parallel lines. Substitution is applied to show that QS is equal to TS, and we conclude the proof by establishing congruence between triangle QSR and triangle TSU using the Angle-Side-Angle (ASA) congruence criterion. Finally, we state the corresponding parts of congruent triangles to confirm that ∠QSR is congruent to ∠TSU.