Midpoint Theorem |AAS & SAS Criterion of Congruency Prove with Diagram

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Theorem: The line segment joining the midpoints of two sides of a
triangle is parallel to the third side and equal to half of it.

Given: A triangle PQR in which S and T are the midpoint of
PQ and PR respectively.

To prove: ST ∥ QR and ST = (frac{1}{2})QR

Construction: Draw RU ∥ QP such that RU meets ST produced at U.
Join SR.

Proof:

           Statement

            Reason

1. In ∆PST and ∆RUT,

(i) PT = TR

(ii) ∠PTS = ∠RTU

(iii) ∠SPT = ∠TRU

1.

(i) T is the midpoint of PR.

(ii) Vertically opposite angles.

(iii) Alternate angles.

2. Therefore, ∆PST ≅ ∆RUT

2. By AAS criterion of congruency.

3. Therefore, PS = RU; ST = TU

3. CPCTC.

4. But PS = QS

4. S is the midpoint of PQ.

5. Therefore, RU = QS and QS ∥ RU.

5. From statements 3, 4 and construction.

6. In ∆SQR and ∆RUS, ∠QSR = ∠URS, QS = RU.

6. From statement 5.

7. SR = SR.

7. Common side

8.  ∆SQR ≅ ∆RUS.

8. SAS criterion of congruency.

9. QR = SU = 2ST and ∠QRS = ∠RSU

9. CPCTC and statement 3.

10. ST = (frac{1}{2})QR and ST ∥ QR

10. By statement 9.

9th Grade Math

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