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

From Midsegment Theorem on Trapezium to HOME PAGE

New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Share this page: What’s this?

Recent Articles

  1. 4th Grade Mental Math on Roman Numerals | Roman Numerals Quiz

    Feb 23, 24 03:55 PM

    In 4th grade mental math on numbers, students can practice different questions on write the Hindu-Arabic numerals, write the Roman Numerals, comparison of roman numerals, addition of roman numerals.

    Read More

  2. 4th Grade Mental Math on Numbers | Mental Math 4th Grade with Answers

    Feb 23, 24 02:24 PM

    4th Grade Mental Math on Numbers
    In 4th grade mental math on numbers, students can practice different questions on numbers in figures, number name, place value, face value, comparison of number and formation of greatest and smallest…

    Read More

  3. Roman Numerals | System of Numbers | Symbol of Roman Numerals |Numbers

    Feb 23, 24 01:28 PM

    List of Roman Numerals Chart
    How to read and write roman numerals? Hundreds of year ago, the Romans had a system of numbers which had only seven symbols. Each symbol had a different value and there was no symbol for 0. The symbol…

    Read More

  4. Worksheet on Roman Numerals |Roman Numerals|Symbols for Roman Numerals

    Feb 22, 24 04:15 PM

    Roman Numbers Table
    Practice the worksheet on roman numerals or numbers. This sheet will encourage the students to practice about the symbols for roman numerals and their values. Write the number for the following: (a) V…

    Read More

  5. Roman Symbols | What are Roman Numbers? | Roman Numeration System

    Feb 22, 24 02:30 PM

    Roman Numbers
    Do we know from where Roman symbols came? In Rome, people wanted to use their own symbols to express various numbers. These symbols, used by Romans, are known as Roman symbols, Romans used only seven…

    Read More