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Math Labs with Activity - Cyclic Quadrilateral

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Math Labs with Activity – Cyclic Quadrilateral

OBJECTIVE

To verify that

  1. the opposite angles of a cyclic quadrilateral are supplementary, and
  2. if one side of a cyclic quadrilateral is produced then the exterior angle is equal to the opposite interior angle.

Materials Required

  1. A piece of cardboard
  2. A sheet of white paper
  3. A sheet of glazed paper
  4. A sheet of tracing paper
  5. A geometry box
  6. A pair of scissors
  7. A tube of glue

Theory
A quadrilateral whose all the four vertices lie on the circumference of the same circle is called a cyclic quadrilateral. In a cyclic quadrilateral, the opposite angles are supplementary and the exterior angle (formed by producing a side) is equal to the opposite interior angle.

Procedure
Step 1: Paste the sheet of white paper on the cardboard.
Draw a circle of any radius on this sheet and also draw a cyclic quadrilateral ABCD.
Step 2: Produce the side CD of the cyclic quadrilateral ABCD to E. Mark the exterior angle ∠ADE (see Figure 29.1).
Math Labs with Activity - Cyclic Quadrilateral 1
Step 3: Make an exact replica of the quadrilateral ABCD on a glazed paper using the tracing paper. Also, make a replica of ∠ADE on the tracing paper.
Step 4: Cut out the quadrilateral ABCD made on the sheet of glazed paper and further cut it into four parts such that each part contains one of the angles ∠A, ∠B, ∠C and ∠D (see Figure 29.2)
Math Labs with Activity - Cyclic Quadrilateral 2
Step 5: Place ∠A and ∠C adjacent to each other (as in Figure 29.3).’What do you observe?
Math Labs with Activity - Cyclic Quadrilateral 3
Step 6: Place ∠B and ∠D adjacent to each other (as in Figure 29.4). What do you observe?
Math Labs with Activity - Cyclic Quadrilateral 4
Step 7: Place the replica of ∠ADE (which you have on the tracing paper) on ∠B of the quadrilateral ABCD on the sheet of white paper. What do you observe?

Observations

  1. When ∠A and ∠C are placed adjacent to each other, they form a linear pair, i.e., ∠A + ∠C = 180°.
  2. When ∠B and ∠D are placed adjacent to each other, they form a linear pair, i.e., ∠B + ∠D = 180°.
  3. When ∠ADE is placed over ∠B, it covers ∠B completely, i.e., ∠ADE = ∠B.

Result
It is verified that

  1. the opposite angles of a cyclic quadrilateral are supplementary, and
  2. if one side of a cyclic quadrilateral is produced then the exterior angle is equal to the opposite interior angle.

Remarks:
The above results are true only for a cyclic quadrilateral.

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