Quadrilaterals
About 7 min · Mathematical Reasoning · English
Time worked out from the lesson's 462 words at 150 words a minute, plus a minute for each question.
Quadrilaterals form a critical bridge between basic geometry and advanced problem-solving in SOF IMO. This topic can appear in both the Mathematical Reasoning section and in Achievers Section problems that combine properties with area calculations or coordinate geometry. Understanding quadrilaterals means mastering the hierarchy: every square is a rectangle, every rectangle is a parallelogram, but not vice versa. The mid-point theorem, which connects triangles to quadrilaterals, is a powerful tool for construction and proof problems.
Key ideas
- Quadrilateral basics: A closed figure with four sides, four vertices and four angles. Sum of all interior angles is always 360°. Quadrilaterals are classified by their side lengths, angle measures and diagonal properties.
- Parallelogram: Opposite sides are parallel and equal. Opposite angles are equal. Consecutive angles are supplementary (sum = 180°). Diagonals bisect each other but are not necessarily equal or perpendicular.
- Rectangle: A parallelogram with all four angles equal to 90°. Diagonals are equal in length and bisect each other, but they are not perpendicular unless the rectangle is a square.
- Rhombus: A parallelogram with all four sides equal. Diagonals are perpendicular bisectors of each other. Opposite angles are equal, and diagonals bisect the vertex angles.
- Square: A rectangle with all sides equal, or equivalently a rhombus with all angles 90°. Diagonals are equal, perpendicular and bisect each other at right angles. Each diagonal bisects opposite angles into 45° each.
Watch out for
- Confusing diagonal properties → Students mix up which quadrilateral has equal diagonals versus perpendicular diagonals. Fix: Remember: Rectangle has equal diagonals; Rhombus and Square have perpendicular diagonals; Square has both equal and perpendicular.
- Assuming all parallelograms have equal diagonals → Only rectangles and squares (special parallelograms) have equal diagonals. Fix: General parallelograms only have diagonals that bisect each other, not equal length.
- Wrong angle calculation in parallelograms → Adding opposite angles instead of consecutive ones to get 180°. Fix: Opposite angles are equal; consecutive angles are supplementary. If angle A = 70°, then angle B = 110°, not 70°.
Quick reference
- Parallelogram: Opposite sides parallel and equal; diagonals bisect each other.
- Rectangle: Parallelogram with 90° angles; diagonals equal and bisect.
- Rhombus: Parallelogram with all sides equal; diagonals perpendicular and bisect.
- Square: All sides equal, all angles 90°; diagonals equal, perpendicular, bisect.
- Angle sum of any quadrilateral = 360°.
Check yourself: 3 questions
1. In a parallelogram ABCD, if angle A = 65°, then what is the measure of angle C?
2. The diagonals of a rhombus are 16 cm and 12 cm. What is the length of each side of the rhombus?
3. In triangle ABC, D and E are the mid-points of sides AB and AC respectively. If BC = 10 cm and DE is parallel to BC, what is the length of DE?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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