Example · angles in a parallelogram
If one angle is 70°, each adjacent angle is 110° and the opposite angle is 70°. The four angles total 360°.
Class 9 › Mathematics › CBSE topic 10
CBSE Mathematics 2026–27 · Topic 10
4-gons (Quadrilaterals)
Understand the idea, practise the reasoning and apply it in an investigation. These original notes follow the regular Class IX Mathematics syllabus; the Advanced course is separate.
What to learn
Be able to explain the key result, choose a method that fits the question, show reasoning clearly and verify the answer.
Core concepts
- A diagonal divides a quadrilateral into two triangles, so its interior angles sum to 360°.
- In a parallelogram, opposite sides and angles are equal, adjacent angles are supplementary, and diagonals bisect each other.
- A rectangle has right angles; a rhombus has equal sides; a square has both. Use properties to classify a figure and justify the classification.
Parallelograms, midpoints, and symmetry
A quadrilateral has four sides and four interior angles whose measures sum to 360°. A parallelogram has both pairs of opposite sides parallel. Its opposite sides are equal, its opposite angles are equal, its adjacent angles are supplementary, and its diagonals bisect each other. Each statement can be proved by drawing a diagonal and comparing the triangles formed using alternate interior angles.
Several tests can establish that a quadrilateral is a parallelogram: both pairs of opposite sides are parallel; both pairs are equal; one pair is both equal and parallel; or the diagonals bisect one another. Use a test only after checking all of its conditions. A rectangle adds four right angles, a rhombus has four equal sides, and a square has both properties.
Midpoint theorem: In a triangle, the segment joining the midpoints of two sides is parallel to the third side and has half its length. The converse says that a line through the midpoint of one side and parallel to a second side bisects the third side. A median joins a vertex to the opposite midpoint; the three medians meet at one point, and that point divides each median in the ratio 2:1 measured from the vertex.
Joining the midpoints of the four sides of any quadrilateral creates a parallelogram: each pair of opposite midpoint-joining segments is parallel to the same diagonal of the original quadrilateral by the midpoint theorem. This is a useful example of proving a new figure’s properties from a simpler triangle result.
Parallelograms have central symmetry: a half-turn (180° rotation) about the intersection of the diagonals maps the figure onto itself. Drawing, folding, and checking coordinates can suggest these properties; a proof explains why they hold for every allowed parallelogram.
Original CBSE-style solved examples
Attempt each calculation before reading the solution.
Practice questions with answers
1. Three angles are 85°, 95° and 110°. Find the fourth angle.
360°−290°=70°.
2. One half of a parallelogram diagonal is 4.5 cm because the diagonals bisect each other. Find the whole diagonal.
9 cm.
Practical investigation
Compare a rectangle, rhombus and square by measuring sides, angles and diagonals. Make a property table and identify shared properties.
Keep a record of materials, steps, observations and conclusion.
Self-check
- Can I state the definition or theorem precisely?
- Can I explain why the method applies?
- Did I check signs, units, labels and the final result?
Use your prescribed NCERT textbook and teacher’s sequence. These are original study notes, not copied textbook answers.