Exercise 8.1 — Quadrilateral Properties

Problems based on properties of quadrilaterals.

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Exercise 8.1 — Quadrilaterals

Exercise 8.1 from Chapter 8, Quadrilaterals, of Class 9 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) covers the classification of quadrilaterals, their properties, and formal proofs involving parallelograms, trapeziums, rectangles, rhombuses, and squares. This exercise combines conceptual true/false questions, a detailed properties comparison table, a geometric proof about an isosceles trapezium, an angle-ratio problem, and a question on the nature of a triangle formed inside a rectangle.

A quadrilateral is any closed polygon with exactly four sides, four angles, and four vertices. The sum of its interior angles is always 360° — a key fact used throughout this exercise.

Quadrilateral Hierarchy Properties Table Isosceles Trapezium Proof Angle Ratio Problem

💡 Foundation fact: The sum of all interior angles of any quadrilateral = 360°. Every formula and property in this chapter builds on this one fact.

The Quadrilateral Family — Hierarchy and Relationships

Understanding which shape is a special case of another is crucial for answering the true/false questions and the properties table in this exercise. The hierarchy flows from the most general to the most specific:

📌 Reading the hierarchy: Every shape higher in the diagram includes all shapes below it as special cases. A square is a rectangle AND a rhombus. A rectangle is a parallelogram. A parallelogram is a trapezium. But the reverse is NOT always true — not every trapezium is a parallelogram.

Question 1 — True or False: Quadrilateral Statements

These six statements test your understanding of the quadrilateral hierarchy. Use the diagram above as your guide — a statement is true only if the first shape is always a special case of (or belongs to) the second shape.

#	Statement	Answer	Reason
(i)	Every parallelogram is a trapezium.	True	A parallelogram has two pairs of parallel sides, which means it certainly satisfies the condition of having at least one pair — so it qualifies as a trapezium.
(ii)	All parallelograms are quadrilaterals.	True	Every parallelogram has exactly four sides, so it is by definition a quadrilateral.
(iii)	All trapeziums are parallelograms.	False	A trapezium has only one pair of parallel sides, while a parallelogram needs two pairs. Not every trapezium has its second pair of sides parallel.
(iv)	A square is a rhombus.	True	A square has all four sides equal AND all angles 90°. A rhombus only requires all sides to be equal. Since a square satisfies that condition, every square is a rhombus.
(v)	Every rhombus is a square.	False	A rhombus has equal sides but its angles need not be 90°. A square also requires all angles to be right angles — which a general rhombus may not have.
(vi)	All parallelograms are rectangles.	False	A rectangle needs at least one right angle (which makes all four 90°). A general parallelogram may have any angle — it does not have to be 90°.

⚠️ Most common error: Students often mark (iii) as True thinking "trapezium is a type of parallelogram." It's exactly the other way around — a parallelogram is a special trapezium, not vice versa.

Question 2 — Properties of Quadrilaterals (YES / NO Table)

This is one of the most important reference tables in the Quadrilaterals chapter. It compares five types of quadrilaterals across ten geometric properties — a thorough understanding of this table will help you answer almost any question about quadrilaterals in your board exam.

Property	Trapezium	Parallelogram	Rhombus	Rectangle	Square
a. Only one pair of opposite sides parallel	YES	NO	NO	NO	NO
b. Two pairs of opposite sides parallel	NO	YES	YES	YES	YES
c. Opposite sides are equal	NO	YES	YES	YES	YES
d. Opposite angles are equal	NO	YES	YES	YES	YES
e. Consecutive angles are supplementary	NO	YES	YES	YES	YES
f. Diagonals bisect each other	NO	YES	YES	YES	YES
g. Diagonals are equal	NO	NO	NO	YES	YES
h. All sides are equal	NO	NO	YES	NO	YES
i. Each angle is a right angle	NO	NO	NO	YES	YES
j. Diagonals are perpendicular to each other	NO	NO	YES	NO	YES

💡 Memory shortcut for the Square column: A square answers YES to every single property — it is the most "complete" quadrilateral, combining all properties of both the rectangle (equal diagonals, right angles) and the rhombus (equal sides, perpendicular diagonals).

📌 Exam-ready patterns to memorize:

Equal diagonals → only Rectangle and Square (not Rhombus).
Perpendicular diagonals → only Rhombus and Square (not Rectangle).
All sides equal → only Rhombus and Square (not Rectangle).
All angles 90° → only Rectangle and Square (not Rhombus).

Question 3 — Proof: In Isosceles Trapezium ABCD, ∠A = ∠B and ∠C = ∠D

In trapezium ABCD, AB ∥ CD and the non-parallel sides (legs) are equal: AD = BC. Such a trapezium is called an isosceles trapezium. This question asks you to prove that the base angles are equal: ∠A = ∠B and ∠C = ∠D.

Isosceles trapezium ABCD with perpendiculars DF and CE

Solution

Prove ∠A = ∠B and ∠C = ∠D

Construction: Draw perpendiculars CE and DF from C and D onto AB, meeting AB at E and F respectively.

Step 1 — In △AFD and △BEC: ∠AFD = ∠BEC = 90° (perpendicular construction) AD = BC (given — isosceles trapezium) DF = CE (perpendicular distance between parallel lines AB ∥ DC is constant) By RHS congruence: △AFD ≅ △BEC ∴ ∠A = ∠B (CPCT) Step 2 — Find ∠C = ∠D: Since AB ∥ DC, co-interior angles give: ∠A + ∠D = 180° and ∠B + ∠C = 180° So: ∠A + ∠D = ∠B + ∠C Since ∠A = ∠B, substitute: ∠B + ∠D = ∠B + ∠C ∴ ∠D = ∠C ✓

✅ Proof strategy recap: Use the RHS congruence rule (Right angle – Hypotenuse – Side) to prove the two right-angled triangles congruent, then use CPCT (Corresponding Parts of Congruent Triangles) to conclude ∠A = ∠B, and finally use the co-interior angle property of parallel lines for ∠C = ∠D.

Question 4 — Angles of a Quadrilateral in Ratio 1 : 2 : 3 : 4

The four angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Since the total angle sum is always 360°, we divide 360° in proportion to these four parts.

Solution

Find each angle when angles are in ratio 1 : 2 : 3 : 4

Sum of ratio parts = 1 + 2 + 3 + 4 = 10 Total angle sum of a quadrilateral = 360° Value of one part = 360° ÷ 10 = 36° 1st angle = 1 × 36° = 36° 2nd angle = 2 × 36° = 72° 3rd angle = 3 × 36° = 108° 4th angle = 4 × 36° = 144° Verification: 36° + 72° + 108° + 144° = 360° ✓

Ratio: 1

36°

1st angle

Ratio: 2

72°

2nd angle

Ratio: 3

108°

3rd angle

Ratio: 4

144°

4th angle

💡 Method shortcut: For any ratio problem involving angles of a quadrilateral: divide 360° by the sum of ratio parts to get the value of "one part," then multiply each ratio number by that value. Always verify by checking all angles add to 360°.

Question 5 — Nature of △ACD Formed by Diagonal of a Rectangle

In rectangle ABCD, diagonal AC is drawn. What type of triangle is △ACD?

Rectangle ABCD with diagonal AC forming △ACD

Solution

Find the nature of △ACD

In rectangle ABCD, every angle = 90°. So ∠D = 90° (angle of the rectangle at vertex D) In △ACD, ∠ADC = 90° A triangle with one angle = 90° is a right-angled triangle. Also: ∠CAD + ∠ACD = 90° (since ∠D = 90°, the other two angles together = 90°)

✅ Conclusion: △ACD is a right-angled triangle, with the right angle at vertex D. The hypotenuse of this triangle is the diagonal AC of the rectangle.

📌 Extension — going further: In a rectangle, both diagonals are equal in length. The diagonal AC is the hypotenuse of the right-angled triangles formed. By the Pythagorean theorem: AC² = AD² + DC² (using the sides of the rectangle as legs).

Common Mistakes to Avoid

Reversing the hierarchy: "Every parallelogram is a trapezium" is TRUE, but "every trapezium is a parallelogram" is FALSE — the relationship works only in one direction.
Confusing rhombus and rectangle properties: A rhombus has perpendicular diagonals but NOT necessarily equal ones. A rectangle has equal diagonals but NOT necessarily perpendicular ones. Only a square has both.
Forgetting to verify angle sums: In ratio problems (Q4), always add your four angles at the end to confirm they total 360°.
Wrong congruence rule in Q3: The proof uses RHS (Right angle – Hypotenuse – Side), not SAS or ASA, because you have a right angle, the hypotenuse (AD = BC), and another side (DF = CE).
Calling △ACD "equilateral" or "isosceles" without reason: The nature of △ACD depends on the dimensions of the rectangle. The only thing always guaranteed is that it is right-angled at D.

⛔ High-risk exam trap: The properties table question often appears in board exams with subtle twists — for example, "Do the diagonals of a rhombus bisect each other?" (YES — all parallelograms' diagonals bisect each other) versus "Are the diagonals of a rhombus equal?" (NO — only rectangles and squares have equal diagonals).

Quick Reference — All Answers at a Glance

Question	Topic	Answer / Result
Q1(i)	Every parallelogram is a trapezium	True
Q1(ii)	All parallelograms are quadrilaterals	True
Q1(iii)	All trapeziums are parallelograms	False
Q1(iv)	A square is a rhombus	True
Q1(v)	Every rhombus is a square	False
Q1(vi)	All parallelograms are rectangles	False
Q2	Properties table (10 properties × 5 shapes)	See table above
Q3	Isosceles trapezium angle proof	∠A = ∠B and ∠C = ∠D (via RHS + CPCT)
Q4	Angles in ratio 1:2:3:4	36°, 72°, 108°, 144°
Q5	Nature of △ACD in rectangle ABCD	Right-angled triangle (∠D = 90°)

What This Exercise Prepares You For

Exercise 8.1 builds the classification and property knowledge you'll need for the rest of Chapter 8, where formal theorems about parallelograms are proved — such as "opposite sides of a parallelogram are equal" and "the diagonals of a parallelogram bisect each other." A thorough understanding of the hierarchy and property table here makes those proofs much more straightforward.

The proof technique from Question 3 (drawing perpendiculars and using RHS congruence) reappears frequently in the Triangles chapter, while the angle-ratio method from Question 4 connects directly to problems in Coordinate Geometry involving angle-based conditions. For Class 8 revision, this chapter also connects to Construction of Quadrilaterals.

📐 Board Exam Tip (Telangana & AP): The properties table (Question 2) is one of the most frequently examined topics in this chapter — it often appears as a 4-mark or 5-mark question where the entire table must be filled in correctly. Memorize the four "special" YES cells: equal diagonals (Rectangle, Square), perpendicular diagonals (Rhombus, Square), all sides equal (Rhombus, Square), and all angles 90° (Rectangle, Square).