Midsegment Theorem on Trapezium

Here we will prove that the line segment joining the midpoints of the nonparallel sides of a trapezium is half the sum of the lengths of the parallel sides and is also parallel to them.


Given: PQRS is a trapezium in which PQ ∥ RS. U and V are the midpoints of QR and PS respectively.

Midsegment Theorem on Trapezium

To prove: (i) UV ∥ RS.

(ii) UV = \(\frac{1}{2}\)(PQ + RS).

Construction: Join QV and produce it to meet RS produced at T.




1. In ∆PQV and ∆STV,

(i) PV = VS.

(ii) ∠PVQ = ∠TVS.

(iii) ∠QPV = ∠VST.


(i) Given.

(ii) Vertically opposite angles.

(iii) Alternate angles.

2. Therefore, ∆PQV ≅ ∆STV.

2. By ASA criterion of congruency.

3. Therefore, PQ = ST.


4. QV = VT.


5. In ∆QRT,

(i) U is the midpoint of QR.

(ii) V is the midpoint of QT.


(i) Given.

(ii) From statement 4.

6. Therefore, UV ∥ RT and UV = \(\frac{1}{2}\)RT.

6. By the Midpoint Theorem.

7. Therefore, UV = \(\frac{1}{2}\)(RS+ ST).

7. From statement 6.

8. UV = \(\frac{1}{2}\)(RS+ PQ).

8. Using statement 3 in statement 7.

9. Therefore, UV ∥ RS and UV = \(\frac{1}{2}\)(PQ+ RS). (Proved)

9. From statement 6 and 8.

9th Grade Math

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