NCERT Book Class 9 Maths Chapter 8 Quadrilaterals

104 MATHEMATICS

CHAPTER 8

QUADRILATERALS

8.1 Properties of a Parallelogram
You have already studied quadrilaterals and their types in Class VIII. A quadrilateral
has four sides, four angles and four vertices. A parallelogram is a quadrilateral in
which both pairs of opposite sides are parallel.
Let us perform an activity.
Cut out a parallelogram from a sheet of paper
and cut it along a diagonal (see Fig. 8.1). You obtain
two triangles. What can you say about these
triangles?
Place one triangle over the other. Turn one around,
if necessary. What do you observe?
Observe that the two triangles are congruent to Fig. 8.1
each other.
Repeat this activity with some more parallelograms. Each time you will observe
that each diagonal divides the parallelogram into two congruent triangles.
Let us now prove this result.
Theorem 8.1 : A diagonal of a parallelogram divides it into two congruent
triangles.
Proof : Let ABCD be a parallelogram and AC be a diagonal (see Fig. 8.2). Observe
that the diagonal AC divides parallelogram ABCD into two triangles, namely, ∆ ABC
and ∆ CDA. We need to prove that these triangles are congruent.