Quadrilaterals — Class 9 IMO

Angle Sum and the Parallelogram

What is the Angle Sum Property of a Quadrilateral?

A quadrilateral is any closed flat geometric shape that has exactly four straight sides, four vertices (corners), and four interior angles. The word comes from "quad" meaning four and "lateral" meaning sides. Everyday examples include a square chessboard, a rectangular smartphone screen, a kite soaring in the sky, or a diamond shape on a playing card.

The angle sum property states that the sum of all four interior angles of any quadrilateral is always exactly 360°. It does not matter if the quadrilateral is perfectly symmetrical or highly irregular; its four corners will always add up to 360°.

To understand why this is true, imagine cutting any quadrilateral into two pieces by drawing a straight line from one corner to the opposite corner. This dividing line is called a diagonal.

Drawing one diagonal splits the quadrilateral cleanly into two separate triangles.
We already know from our study of triangles that the angles inside any single triangle always add up to 180°.
Since our quadrilateral is now made up of exactly two triangles, the total sum of all the angles is simply the sum of the angles of both triangles: 180° + 180° = 360°.

This rule allows us to easily calculate the measurement of a missing fourth angle if we already know the measurements of the other three angles.

Example 1: Three angles of a quadrilateral are 80°, 90° and 100°. The fourth?

360° − (80° + 90° + 100°) = 90°.

Example 2: One angle of a parallelogram is 70°. Its opposite angle?

Also 70° — opposite angles of a parallelogram are equal.

Quick recap

Quadrilateral angles sum to 360°.
Parallelogram: opposite sides/angles equal, diagonals bisect each other.

✓ Quick check

ABCD is a parallelogram. If a line PQ is drawn parallel to diagonal AC cutting AB at P and BC at Q, then triangle APQ is equal in area to which of the following triangles?

Triangles APQ and CPQ share the same base PQ and lie between the same parallel lines PQ and AC. Therefore, their areas are equal.

In a parallelogram ABCD, if angle A is a right angle, then the parallelogram is specifically classified as a:

By definition, a parallelogram with one right angle is a rectangle (since opposite and adjacent angles will also become 90°).

Special Parallelograms

What are the Different Types of Quadrilaterals?

While all quadrilaterals share the property of having four sides and angles adding up to 360°, they can look completely different based on how their sides and angles relate to each other. We classify quadrilaterals into specific families using traits like parallel sides, equal side lengths, and right angles.

Let us explore the special members of the quadrilateral family:

Trapezium: A quadrilateral that has exactly one pair of parallel sides.
Parallelogram: A quadrilateral where both pairs of opposite sides are parallel. This shape acts as the parent to several other special shapes because it passes down its unique mathematical features.
Rhombus: A special type of parallelogram where all four sides are equal in length. Think of a diamond shape or a slanted square.
Rectangle: A special type of parallelogram where all four interior angles are right angles (90°).
Square: The most perfect, highly regular quadrilateral. It is a parallelogram that is both a rhombus and a rectangle, meaning all four sides are equal and all four angles are 90°.
Kite: A quadrilateral that has two pairs of equal-length adjacent (neighboring) sides, but opposite sides are not equal.

Let us compare the structural traits of these shapes in a comprehensive reference table:

Quadrilateral Type	Opposite Sides Parallel?	Opposite Sides Equal?	All Angles = 90°?	Diagonals Behavior
Parallelogram	Both pairs parallel	Yes, opposite equal	No	Bisect each other
Rhombus	Both pairs parallel	Yes, all 4 equal	No	Bisect at 90°
Rectangle	Both pairs parallel	Yes, opposite equal	Yes, all 90°	Equal length and bisect
Square	Both pairs parallel	Yes, all 4 equal	Yes, all 90°	Equal length and bisect at 90°
Kite	No	Adjacent pairs equal	No	Intersect at 90°

Example 1: What is special about a rhombus's diagonals?

They bisect each other at right angles (but are not equal).

Example 2: A parallelogram has equal diagonals. What can it be?

A rectangle (or a square, which is a special rectangle).

Quick recap

Rectangle: equal diagonals; rhombus: perpendicular diagonals.
Square: diagonals equal AND perpendicular bisectors.

✓ Quick check

In a rectangle ABCD, diagonal AC = 25 cm and side AB = 24 cm. Find the perimeter of the rectangle.

In a rectangle, angle B = 90°. In right-angled triangle ABC, BC = √(AC² − AB²) = √(25² − 24²) = √(625 − 576) = √49 = 7 cm. Perimeter of the rectangle = 2(length + width) = 2(24 + 7) = 2 × 31 = 62 cm.

In a triangle PQR, S and T are the mid-points of PQ and PR respectively. If QR = 2x + 4 and ST = x + 2, which statement is true about the values?

By mid-point theorem, ST = QR/2. Substituting the expressions: x + 2 = (2x + 4)/2 = x + 2. Since this simplifies to an identity (x+2 = x+2), it is true for any valid x that keeps side lengths positive.

Mid-point Theorem and Tests for a Parallelogram

What is the Mid-Point Theorem?

The Mid-Point Theorem is a powerful geometric rule that connects the mid-points of the sides of a triangle. A mid-point is a point located exactly halfway along a line segment, splitting it into two equal halves.

The theorem states: The line segment connecting the mid-points of any two sides of a triangle is automatically parallel to the third side, and its length is exactly half the length of that third side.

Imagine you are looking at a large triangular tent. If you find the exact halfway point on the left support pole and the exact halfway point on the right support pole, and connect them with a horizontal beam:

That new beam will run perfectly parallel to the ground (the base of the tent).
The length of that beam will be exactly half as wide as the base of the tent.

The converse of the mid-point theorem is also true: If you start at the mid-point of one side of a triangle and draw a line that runs perfectly parallel to the base line, it will hit the opposite side at its exact mid-point.

This theorem is highly useful for proving relationships in more complex geometric structures, like showing that joining the mid-points of any random quadrilateral creates a perfect parallelogram!

Example 1: Midpoints of two sides joined; third side is 12 cm. Segment length?

Half of 12 = 6 cm, and it is parallel to that side.

Example 2: A quadrilateral has one pair of sides equal and parallel. What is it?

A parallelogram.

Quick recap

Mid-point segment is parallel to and half the third side.
One pair of sides equal AND parallel ⇒ parallelogram.

✓ Quick check

If ABCD is a parallelogram, which of the following expressions correctly represents the relationship between its consecutive interior angles?

In a parallelogram, consecutive interior angles are supplementary because the opposite sides are parallel. Therefore, angle A + angle B = 180°.

If the mid-points of a rhombus sides are connected in order, the resulting quadrilateral is always a:

The diagonals of a rhombus are perpendicular. The sides of the inner quadrilateral formed by joining the mid-points are parallel to these perpendicular diagonals, meaning the corner angles of the inner figure are 90°. Thus, it forms a rectangle.