Revision and important questions: Quadrilaterals

Quadrilaterals22 min readLesson 6 of 7

6.1 Property table Quadrilateral Definition Opposite sides Angles Diagonals Parallelogram both pairs of opposite sides parallel equal and parallel opposite equal, adjacent supplementary bisect each other Rectangle parallelogram with all angles 90° equal and parallel all 90° equal and bisect each other Rhombus parallelogram with all sides equal all four equal, opposite parallel opposite equal, adjacent supplementary bisect each other at 90° and bisect angles Square rectangle and rhombus together all equal and parallel all 90° equal, bisect at 90°, bisect angles Kite two pairs of adjacent equal sides not both pairs equal the angles between equal sides are equal the diagonal between the equal pairs bisects the other at 90° Trapezium at least one pair of parallel sides one pair parallel co-interior angles on parallel sides add to 180° no general rule Isosceles trapezium trapezium with equal non-parallel sides one pair parallel two pairs of equal base angles equal diagonals

Quadrilateral	Definition	Opposite sides	Angles	Diagonals
Parallelogram	both pairs of opposite sides parallel	equal and parallel	opposite equal, adjacent supplementary	bisect each other
Rectangle	parallelogram with all angles 90°	equal and parallel	all 90°	equal and bisect each other
Rhombus	parallelogram with all sides equal	all four equal, opposite parallel	opposite equal, adjacent supplementary	bisect each other at 90° and bisect angles
Square	rectangle and rhombus together	all equal and parallel	all 90°	equal, bisect at 90°, bisect angles
Kite	two pairs of adjacent equal sides	not both pairs equal	the angles between equal sides are equal	the diagonal between the equal pairs bisects the other at 90°
Trapezium	at least one pair of parallel sides	one pair parallel	co-interior angles on parallel sides add to 180°	no general rule
Isosceles trapezium	trapezium with equal non-parallel sides	one pair parallel	two pairs of equal base angles	equal diagonals

6.2 Formula box and essential facts Angle sum of a quadrilateral is 360°. In a parallelogram, adjacent angles are supplementary and opposite angles are equal. In a rectangle, the diagonals are equal and bisect each other. In a square, the diagonals are equal, bisect each other at 90°, and bisect the angles of the square. In a rhombus, the diagonals bisect each other at 90° and bisect the angles. In an isosceles trapezium, the angles on the same parallel side are supplementary, and the base angles are equal. A quadrilateral with equal diagonals that bisect each other is a rectangle; if the diagonals also meet at 90°, it is a square.

∠ A + ∠ B + ∠ C + ∠ D = 36 0^{\circ}

Angle sum of a quadrilateral

∠ A + ∠ B = 18 0^{\circ}

Adjacent angles in a parallelogram

6.3 Ten important questions Question 1. Three angles of a quadrilateral are 75°, 85° and 100°. Find the fourth. Answer. 360 - (75 + 85 + 100) = 360 - 260 = 100°. Question 2. The angles of a quadrilateral are in the ratio 2:3:4:6. Find them. Answer. The ratio parts add to 2 + 3 + 4 + 6 = 15. Each part is 360/15 = 24°. The angles are 48°, 72°, 96° and 144°. Check: 48 + 72 + 96 + 144 = 360. Question 3. In parallelogram ABCD, ∠A = 65°. Find all the angles. Answer. Opposite angles are equal, so ∠C = 65°. Adjacent angles are supplementary, so ∠B = ∠D = 180 - 65 = 115°. Question 4. The adjacent angles of a parallelogram are in the ratio 2:3. Find them. Answer. Adjacent angles add to 180°, so the parts are 180/5 = 36°. The angles are 72° and 108°. Hence the parallelogram has angles 72°, 108°, 72° and 108°. Question 5. In rhombus ABCD, ∠A = 72°. Find the other angles and the angles made by diagonal BD with AB and AD. Answer. Opposite angles are equal, so ∠C = 72°. Adjacent angles are supplementary, so ∠B = ∠D = 108°. The diagonal BD bisects the angles at B and D, so it makes 54° angles with BA and BC, and with DA and DC. At A, the diagonal AC would bisect the 72° angle into 36° and 36°.

Question 6. In rectangle ABCD, the diagonal AC makes 35° with AB. Find ∠CAB, ∠ACB and ∠ACD. Answer. ∠CAB = 35°. Triangle ABC is right angled at B, so ∠ACB = 180 - 90 - 35 = 55°. Since AD is parallel to BC and AC is a transversal, ∠ACD = ∠CAB = 35° (alternate angles). In triangle ACD, the remaining angle ∠CAD = 180 - 90 - 35 = 55°. Question 7. A kite has AB = AD = 5 cm and BC = CD = 8 cm. The diagonal AC is drawn. What can you say about triangles ABC and ADC? Answer. AB = AD, BC = CD and AC is common, so △ABC ≅ △ADC by SSS. Hence ∠B = ∠D and the diagonal AC bisects ∠A and ∠C. The diagonal BD is bisected at right angles by AC. Question 8. In trapezium PQRS with PQ parallel to RS, ∠P = 115° and ∠Q = 95°. Find ∠R and ∠S. Answer. Co-interior angles on the same side of the parallel sides are supplementary. ∠S = 180 - 115 = 65° and ∠R = 180 - 95 = 85°. Check: 115 + 95 + 85 + 65 = 360. Question 9. A quadrilateral has equal diagonals that bisect each other. What is it? What if the diagonals also meet at 90°? Answer. Equal diagonals that bisect each other make a rectangle. If they also meet at 90°, all four sides become equal, so the figure is a square. Question 10. State whether each statement is true or false and justify: (i) Every square is a rhombus. (ii) Every rhombus is a square. (iii) Every rectangle is a parallelogram. (iv) Every parallelogram is a rectangle. (v) A kite can be a parallelogram. Answer. (i) True, since all four sides of a square are equal. (ii) False, a rhombus need not have right angles. (iii) True, a rectangle has both pairs of opposite sides parallel. (iv) False, a parallelogram need not have right angles. (v) True only for a rhombus (and a square), because a kite that is a parallelogram must have all sides equal; a non-rhombus kite is not a parallelogram.

6.4 Common mistakes to avoid Using "opposite sides equal" as the only test for a rectangle; it describes a parallelogram. Forgetting that the diagonal of a rectangle splits it into two congruent right triangles. Treating a square as separate from rectangles and rhombuses instead of as their overlap. Assuming every rhombus is a square, or every kite is a rhombus. Mixing up the co-interior angle pairs in a trapezium. In a construction, forgetting that diagonals of a parallelogram bisect each other and of a rhombus meet at 90°. Writing an answer without the property or congruence that justifies it.

Exam

In an exam, write the property you are using before the calculation. For a "prove that" question, name the triangles, the congruence condition and the corresponding parts. For a construction, list the steps and give the reason for each mark. A diagram without reasons earns fewer marks than a short, labelled proof.

in hinglish

Revision ke liye property table yaad rakho. Har quadrilateral ke angles ka sum 360° hai. Parallelogram mein opposite angles equal aur adjacent angles supplementary hote hain. Rectangle ke diagonals equal hote hain aur bisect karte hain. Rhombus ke diagonals 90° par bisect karte hain aur angles ko bisect karte hain. Square mein rectangle aur rhombus dono ki properties hoti hain. Kite mein adjacent equal pairs aur trapezium mein parallel sides ka co-interior angle rule yaad rakho. Classification questions mein counterexample dhoondho: rhombus square nahi ho sakta, kite rhombus nahi ho sakta, aur parallelogram rectangle nahi ho sakta. Exam mein pehle property likho, phir calculation. "Prove that" questions mein triangles, congruence condition aur corresponding parts ke naam likho. Construction questions mein midpoint, angle aur equal lengths ka reason do. Answers ko check karo: angles ka sum 360° aur opposite sides/angles ki equality se verify karo.