Parallelograms, Rhombus and Trapezium

4.2 Squares: A Special Rectangle

All the quadrilaterals above are rectangles, including (iv). Quadrilateral (iv) is a rectangle because all its angles are 90°. However, it is a special kind of rectangle with all sides of equal length. We know that this quadrilateral is also called a square.

Thus, every square is a rectangle, but clearly every rectangle is not a square. This relation can be pictorially represented using a Venn diagram^?. We have seen these diagrams earlier. In a Venn diagram, a set of objects is represented as points inside a closed curve. Typically, these closed curves are ovals or circles.

4.3 Parallelograms

Is a rectangle a parallelogram? A rectangle has opposite sides parallel. So, it satisfies the parallelogram's definition. Hence, it is indeed a parallelogram. More specifically, a rectangle is a special kind of parallelogram with all its angles equal to 90°.

Constructing a Parallelogram

To understand the relations between the sides and the angles of a parallelogram, let us construct one.

Given: Draw a parallelogram with adjacent sides of lengths 4 cm and 5 cm, and an angle of 30° between them.

Step 1: Draw line segments AB = 4 cm and AD = 5 cm with 30° between them.
Step 2: Draw a line parallel to AB through the point D and a line parallel to AD through B. Mark the point at which these lines intersect as C. ABCD is the required parallelogram.

4.4 Rhombus

Study the following: parallelogram GAME in which all four sides are equal. Thus, every rhombus is a parallelogram, and the properties of a parallelogram hold true for a rhombus as well. The adjacent angles of a rhombus add up to 180°, and the opposite angles are equal.

As opposite sides are parallel, GAME is also a parallelogram. Thus, every rhombus is a parallelogram and the properties of a parallelogram hold true for a rhombus as well. The adjacent angles of a rhombus add up to 180°, and the opposite angles are equal.

Thus, in rhombus ABCD, \(\angle A = \angle C = 50°\), and \(\angle D = \angle B = 180° - 50° = 130°\).

Where will the set of squares occur in this diagram? We know that a square is a rectangle. Since the opposite sides of a square are parallel, a square is also a parallelogram. Further, since all the sides of a square have the same length, a square is also a rhombus. Thus, the Venn diagram will be as follows:

Properties of a Rhombus

1. All the sides of a rhombus are equal to each other.
2. The opposite sides of a rhombus are parallel to each other.
3. In a rhombus, the adjacent angles add up to 180°, and the opposite angles are equal.
4. The diagonals of a rhombus bisect each other at right angles.
5. The diagonals of a rhombus bisect its angles.

4.5 Trapezium

Parallelograms are quadrilaterals that have both pairs of opposite sides parallel. We get a new type of quadrilateral if we relax this condition.

Construct a trapezium. Measure the base angles (marked in the figure). Can you find the remaining angles without measuring them? Since PQ ∥ SR, we have:

Using these facts, the remaining angles can easily be found. Verify your answer through measuring.

When the non-parallel sides of a trapezium have the same lengths, the trapezium is called an isosceles trapezium.

Construct an isosceles trapezium UVWX, with UV ∥ XW. Measure ∠U. Can you find the remaining angles without measuring them? Since XW ∥ UV, ∠XUV + ∠UXW = 90°, and a = 180° − ∠XUV = 90°, and \(b = 180° - \angle WZY = 90°\) (since the internal angles on the same side of a transversal add up to 180°). Hence XWYZ is a rectangle.