**Math Labs with Activity – Verify the Properties of a Trapezium**

**OBJECTIVE**

To verify the properties of a trapezium

**Materials Required**

- Two sheets of white paper
- A geometry box
- A pair of scissors
- A tube of glue

**Theory**

By geometry, we know that in a trapezium ABCD, if AB || DC and E and F are the midpoints of the sides AD and BC respectively then

- EE || AB, and
- EF = ½(AB+DC).

**Procedure**

**Step 1:** Construct a trapezium ABCD (in which AB || DC)

on a sheet of white paper. Mark the midpoints E and F of the non-parallel sides AD and BC respectively (the midpoints of the sides can be obtained by the method of paper folding). Join EE (see Figure 24.1).

**Step 2:** Mark the angles in the diagram as shown in Figure 24.2.

**Step 3:** Paste the two quadrilaterals ABFE and EFCD as shown in Figure 24.3.

**Observations and Calculations**

- In Figure 24.2, for the trapezium ABCD we have AB || DC.

∴ ∠5 + ∠8 = 180° (consecutive interior angles) and so, in Figure 24.3, ∠5 and ∠8 form a linear pair, i.e., AD’ is a straight line.

Also, in Figure 24.2 we have ∠6 + ∠7 = 180° (since ∠6 and ∠7 form a linear pair) and so, in Figure 24.3, ∠6 and ∠7 form a linear pair, i.e., EE’ is a straight line.

Hence, AD’E’E is a quadrilateral. - In Figure 24.2, for the trapezium ABCD we have AB || DC.

∠1 + ∠4 = 180° (consecutive interior angles) and so, in Figure 24.3, AE || E’D’ (since ∠1 and ∠4 are consecutive interior angles). - In Figure 24.2, we have AE = ED (∴ E is the midpoint of AD) in Figure 24.3, we have AE=E’D’ (since E’D’=ED).

Thus, we have AE || E’D’ and AE =E’D’.

Hence, the quadrilateral AD’E’E is a parallelogram (a pair of opposite sides being equal and parallel). - AD’E’E being a parallelogram, we have

(i) EE’ || AD’ and (ii) EE’ = AD’.

Now, EE’ || AD’ => EF || AB (see Figure 24.3)

and EE’=AD’ => EF+E’F’ =AB+C’D’ => 2EF=AB+CD

(∴ E’F’=EF and C’D’=CD) => EF = ½(AB+DC).

**Result**

It is verified that in a trapezium ABCD, if AB || DC, E is the midpoint of side AD and F is the midpoint of side BC then

- EF || AB
- EF = ½ (AB+DC)

Math Labs with ActivityMath LabsMath Lab ManualScience LabsScience Practical Skills

## Leave a Reply