**Math Labs with Activity – Verify the Properties of the Diagonals of a Parallelogram**

**OBJECTIVE**

To verify the properties of the diagonals of a parallelogram

**Materials Required**

- A sheet of white paper
- A sheet of glazed paper
- A geometry box
- A pair of scissors

**Theory**

By geometry, we know that

- a diagonal of a parallelogram divides it into two congruent triangles, and
- the diagonals of a parallelogram bisect each other.

**Procedure**

**Step 1:** Construct a parallelogram ABCD on the sheet of white paper.

**Step 2:** Draw the diagonal AC of the parallelogram.

**Step 3:** Make an exact copy of ΔABC on the glazed paper. Label it as ΔA’B’C’. Cut ΔA’B’C formed on the glazed paper.

**Step 4:** Rotate ΔA’B’C’ formed on the glazed paper and place it over the ΔACD as shown in Figure 21.1. Record your observations (see Observation 1).

**Step 5:** Remove ΔA’B’C’. In the parallelogram ABCD draw the other diagonal BD.

**Step 6:** Mark the point O where the diagonals AC and BD intersect.

**Step 7:** Fold the paper along the line passing through the point O such that the line OA falls over the line OC as shown in Figure 21.2. Record your observations (see Observation 2).

**Step 8:** Fold the paper along the line passing through the point O such that the line OB falls over the line OD as shown in Figure 21.2. Record your observations (see Observation 3).

**Observations**

- We observe that ΔA’B’C’ exactly covers ΔACD.

Therefore, ΔA’B’C’ is congruent to ΔACD, i.e., ΔABC is congruent to ΔACD. - During the first fold, when the line OA falls over the line OC, we observe that the point A falls exactly over the point C. This shows that OA = OC, i.e., point O is the midpoint of the diagonal AC. So, the diagonal BD bisects the diagonal AC.
- During the second fold, when the line OB falls over the line OD, we observe that the point B falls exactly over the point D. This shows that OB = OD, i.e., point O is the midpoint of the diagonal BD. So, the diagonal AC bisects the diagonal BD.

**Result**

It is verified that

- a diagonal of a parallelogram divides it into two congruent triangles, and
- the diagonals of a parallelogram bisect each other.

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