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Quadrilaterals is where Class 9 geometry starts to pay back: once you know the parallelogram tests and the mid-point theorem, a lot of hard-looking problems collapse into two or three lines. This guide covers what the new chapter includes, works four problems with numbers you can check, and lists the traps.
What is in the new Class 9 quadrilaterals chapter?
In NCERT's Ganita Manjari Part II, this is Chapter 12, Quadrilaterals, which calls them 4-gons. It proves the properties of parallelograms and their converses, then uses them for the mid-point theorem and its converse, the centroid theorem, the mid-point theorem for quadrilaterals (the Varignon parallelogram), and tiling the plane with any 4-gon. The CBSE Class IX curriculum for 2026-27 lists "4-gons (Quadrilaterals)" in the Geometry unit (25 of the 80 theory marks), including finding the fourth vertex of a parallelogram from coordinates.
How do you prove a quadrilateral is a parallelogram?
A parallelogram has both pairs of opposite sides parallel. The chapter's Theorem 1 says that in a parallelogram, opposite sides are equal, opposite angles are equal and the diagonals bisect each other. Theorems 2 to 5 give the converses: a quadrilateral is a parallelogram if any one of these holds.
Both pairs of opposite sides are equal.
Both pairs of opposite angles are equal.
The diagonals bisect each other.
One pair of opposite sides is both equal and parallel.
The last test is the quickest in proofs, but it needs the same pair to be equal and parallel. A quadrilateral with one pair of sides parallel and the other pair equal can be an isosceles trapezium, which is not a parallelogram.
Worked example 1: angles of a parallelogram
In parallelogram ABCD, angle A = (3x + 10) degrees and angle B = (2x − 5) degrees. Find all four angles.
Adjacent angles of a parallelogram add up to 180 degrees (co-interior angles, AD parallel to BC).
3x + 10 + 2x − 5 = 180, so 5x = 175 and x = 35.
Angle A = 115 degrees, angle B = 65 degrees, and by opposite angles, angle C = 115 and angle D = 65.
Check: 115 + 65 + 115 + 65 = 360.
What is the mid-point theorem?
In the chapter's words (Theorem 6): "The segment joining the midpoints of two sides of a triangle is parallel to the third side and has half the length." The converse (Theorem 7): a line through the midpoint of one side, parallel to another side, bisects the third side.
Worked example 2: in triangle ABC, D and E are the midpoints of AB and AC, and BC = 14 cm. Then DE is parallel to BC and DE = 7 cm.
Worked example 3: any quadrilateral PQRS has diagonals PR = 10 cm and QS = 16 cm. Join the midpoints of its four sides. What shape do you get, and what is its perimeter?
In triangle PQR, the segment joining the midpoints of PQ and QR is parallel to PR and equals 5 cm (mid-point theorem).
In triangle PSR, the segment joining the midpoints of PS and SR is also parallel to PR and equals 5 cm.
The same argument with diagonal QS gives the other two sides: 8 cm each.
Opposite sides are equal and parallel, so the shape is a parallelogram (the Varignon parallelogram, Theorem 9). Perimeter = 5 + 8 + 5 + 8 = 26 cm.
This works for any quadrilateral, which is what makes it surprising.
How do medians and coordinates fit in?
The centroid theorem (Theorem 8) says the three medians of a triangle meet at one point, which divides each median in the ratio 2:1, the longer part towards the vertex. If median AD is 12 cm, the centroid G gives AG = 8 cm and GD = 4 cm.
Worked example 4: three vertices of parallelogram ABCD are A(1, 2), B(5, 3) and C(7, 7). Find D.
The diagonals bisect each other, so AC and BD have the same midpoint.
Midpoint of AC = ((1 + 7)/2, (2 + 7)/2) = (4, 4.5).
If D = (x, y), the midpoint of BD is ((5 + x)/2, (3 + y)/2) = (4, 4.5). So x = 3 and y = 6.
D = (3, 6). The midpoint method is in our guide to coordinate geometry in Class 9.
What mistakes should you avoid?
Assuming a property you were asked to prove. If the question says "prove ABCD is a parallelogram", you cannot use its diagonals bisecting each other unless you prove that first.
Mixing up a test and a property. "Diagonals of a parallelogram bisect each other" (property) is different from "if diagonals bisect each other, it is a parallelogram" (test).
Dropping the "half". The mid-point theorem gives both parallel and half the length. Use both.
Getting the centroid ratio upside down. The 2 is from vertex to centroid.
A practice routine for this chapter
A review by Dunlosky and colleagues (2013) rated practice testing and distributed practice the most useful study techniques it examined. Apply both:
Write the four parallelogram tests and the mid-point theorem from memory, then check.
Two days later, do worked example 3 with new diagonal lengths, and one fourth-vertex problem.
A week later, prove that the diagonals of a parallelogram bisect each other, using the congruence tests from our guide to congruent triangles in Class 9.
If a proof stalls at "which triangles do I compare?", the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Draw any four-sided shape tonight, join its side midpoints, and check with a ruler that you get a parallelogram.
