Class 9 Maths • Chapter 8 • NCERT Core

Quadrilaterals

Vardaan Learning Institute | School-Exam Focused Notes

📐 1. Angle Sum Property of a Quadrilateral

Theorem 8.1: The sum of the angles of a quadrilateral is 360°.

Proof Idea: Draw a diagonal. It divides the quadrilateral into two triangles. The sum of angles of each triangle is 180°. Total = 180° + 180° = 360°.

🔷 2. Properties of a Parallelogram

A quadrilateral is a parallelogram if its opposite sides are parallel.

📐 Essential Parallelogram Theorems (The 8.X Series)

Theorem 8.1: A diagonal divides a parallelogram into two congruent triangles.
Theorem 8.2: In a parallelogram, opposite sides are equal. (And its converse 8.3: If opposite sides of a quad are equal, it is a parallelogram).
Theorem 8.4: In a parallelogram, opposite angles are equal. (And converse 8.5).
Theorem 8.6: The diagonals of a parallelogram bisect each other. (And converse 8.7).

Conditions for a Quadrilateral to be a Parallelogram

A quadrilateral is a parallelogram if ANY ONE of the following is true:

Both pairs of opposite sides are parallel.
Both pairs of opposite sides are equal.
Both pairs of opposite angles are equal.
The diagonals bisect each other.
Theorem 8.8: A pair of opposite sides is both equal and parallel. (Very powerful for proofs!)

🔲 3. Special Parallelograms (Rectangle, Rhombus, Square)

Shape	Definition	Key Diagonal Properties
Rectangle	A parallelogram with one right angle (all become 90°).	Diagonals are equal and bisect each other.
Rhombus	A parallelogram with all sides equal.	Diagonals bisect each other at right angles (90°).
Square	A rectangle with all sides equal.	Diagonals are equal AND bisect at right angles.

Proof Strategy To prove a rhombus is a square: First prove it is a rhombus, then prove one angle is 90° OR prove its diagonals are equal.

🔗 4. The Mid-Point Theorem (Very Important!)

📐 Theorem 8.9: The Mid-Point Theorem

Statement: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.

If D and E are mid-points of AB and AC in $\triangle ABC$, then:
1. $\mathbf{DE \parallel BC}$
2. $\mathbf{DE = \frac{1}{2} BC}$

📐 Theorem 8.10: Converse of Mid-Point Theorem

The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.
(If D is mid-point of AB, and DE $\parallel$ BC, then E is the mid-point of AC).

✏️ Practice Questions — Quadrilaterals (25 Questions)

Section A — 1 Mark Questions Easy

Q1. Angles of a quadrilateral are in ratio 3:5:9:13. Find all angles.

Q2. If one angle of a parallelogram is 60°, find the other three.

Q3. True/False: Every square is a rectangle.

Q4. True/False: Every rectangle is a square.

Q5. Two consecutive angles of a parallelogram are in ratio 4:5. Find them.

Q6. What is the sum of exterior angles of a quadrilateral?

Q7. Name the quad. whose diagonals bisect each other at 90° but are not equal.

Q8. If diagonals of a parallelogram are equal, what is it?

Section B — Parallelogram Proofs Medium

Q9. In $\triangle ABC$, $D, E, F$ are midpoints of $AB, BC, CA$. Show $\triangle ABC$ is divided into 4 congruent triangles.

Q10. Show that the diagonals of a rhombus are perpendicular to each other.

Q11. Prove that if joining the midpoints of adjacent sides of a quadrilateral, the resulting figure is a parallelogram.

Q12. ABCD is a parallelogram and AP, CQ are perpendiculars from A and C on BD. Prove $\triangle APB \cong \triangle CQD$ and AP = CQ.

Q13. Show that the bisectors of angles of a parallelogram form a rectangle.

Q14. If an angle of a parallelogram is two-thirds its adjacent angle, find angles.

Section C — Essential NCERT Proofs Hard

Q15. Prove that a diagonal of a parallelogram divides it into two congruent triangles.
Q16. ABCD is a rhombus. Show that diagonal AC bisects $\angle A$ as well as $\angle C$.
Q17. In parallelogram ABCD, points P and Q are taken on diagonal BD such that DP = BQ. Prove that APCQ is a parallelogram.
Q18. ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove $ar(ADX) = ar(ACY)$.
Q19. Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a rectangle is a rhombus.
Q20. Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a rhombus is a rectangle.

Section D — Challenge Questions Hard

Q21. In $\triangle ABC$, D, E and F are respectively the mid-points of sides AB, BC and CA. Show that $\triangle ABC$ is divided into four congruent triangles by joining D, E and F.
Q22. A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus.
Q23. Let l, m, n be three parallel lines intersected by transversals p and q. If l, m, n cut off equal intercepts on p (AB=BC), prove they cut off equal intercepts on q (DE=EF). (Intercept Theorem).
Q24. ABC is a triangle right angled at C. A line through mid-point M of hypotenuse AB parallel to BC intersects AC at D. Prove: (i) D is mid-point of AC, (ii) $MD \perp AC$, (iii) $CM = MA = \frac{1}{2}AB$.
Q25. In a parallelogram, show that the ratio of the areas of a triangle to the parallelogram sharing the same base and between same parallels is 1:2.