Hard Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

In this hard level, we'll tackle advanced properties, proofs, and complex problem-solving with quadrilaterals.

Let's master the most challenging concepts in quadrilaterals!

1. Define a kite.

A kite is a quadrilateral with two pairsone pair of adjacentopposite sides equal

Perfect! Kite has two pairs of consecutive/adjacent sides equal.

2. Write one property of diagonals of a parallelogram.

bisectare perpendicular to each other

Correct! They cut each other into two equal parts.

3. Write the number of diagonals in a quadrilateral.

Number of diagonals = 24

Perfect! Every quadrilateral has exactly 2 diagonals.

4. Define a rhombus.

A rhombus is a parallelogramquadrilateral with all fourtwo sides equalunequal

Excellent! Rhombus is an equilateral parallelogram.

5. What is the sum of angles of a quadrilateral?

Sum = °

Correct! This is the fundamental angle sum property.

__{.m-red}6. In a parallelogram, opposite sides are __.

Opposite sides are equalunequal and parallelperpendicular

Perfect! Both properties apply to opposite sides.

7. Write one condition for a parallelogram to be a rectangle.

Condition: All angles must be ° or right anglesacute angles

Excellent! When all angles are 90°, parallelogram becomes rectangle.

8. What is the angle between the diagonals of a square?

Angle = °

Correct! Square diagonals are perpendicular to each other.

9. Write the property of adjacent angles in a parallelogram.

Adjacent angles are supplementarycomplementary

Their sum = °

Perfect! Adjacent angles add up to 180°.

10. Write one difference between a kite and a rhombus.

Kite: AdjacentOpposite pairs of sides equal

Rhombus: AllTwo sides equal

Excellent! Rhombus has all sides equal, kite has adjacent pairs equal.

Drag each property to the correct quadrilateral:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Find the fourth angle of a quadrilateral whose other three angles are 110°, 70°, and 80°.

Sum of three angles = 110 + 70 + 80 = °

Fourth angle = 360 - 260 = °

Perfect! The fourth angle is 100°.

2. The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Find all the angles.

Let angles be 3x, 4x, 5x, 6x

Sum = 3x + 4x + 5x + 6x = 18x18

18x =

x =

Angles: 3×20 = °, 4×20 = °, 5×20 = °, 6×20 = °

Excellent! The angles are 60°, 80°, 100°, and 120°.

3. Write the differences between parallelogram and trapezium.

Parallelogram: BothOne pairs of opposite sides parallel

Trapezium: OneBoth pair of sides parallel

Parallelogram: Opposite sides are equalunequal

Trapezium: No sides necessarily equalparallel

Great! These are the key differences.

4. Prove that the sum of opposite angles of a parallelogram is 180°.

Let angles be A, B, C, D

Adjacent angles are supplementarycomplementary

So A + B = °

Also B + C = °

But opposite angles are equalsupplementary

So A = C and B = D

Therefore: A + C = 2A or 2C (not 180°)

Note: Opposite angles are equal, not supplementary! Adjacent angles are supplementary.

5. If one angle of a rectangle is 90°, what are the remaining three angles?

In a rectangle, all angles are °

So remaining three angles are: °, °, °

Perfect! All four angles in a rectangle are 90°.

6. The diagonals of a square are equal and intersect at 90°. Prove it using properties.

Explanation:

Square is a rectangleparallelogram, so diagonals are equalunequal

Square is also a rhombustrapezium, so diagonals are perpendicularparallel

Therefore, diagonals are both equal and perpendicular

Excellent! Square has properties of both rectangle and rhombus.

7. In a parallelogram ABCD, ∠A = 70°. Find ∠B, ∠C, and ∠D.

Adjacent angles are supplementary

∠A + ∠B = °

∠B = 180 - 70 = °

Opposite angles are equal

∠C = ∠A = °

∠D = ∠B = °

Perfect! Angles are: A=70°, B=110°, C=70°, D=110°.

8. The diagonals of a rhombus are 24 cm and 18 cm. Find its side.

Use Pythagoras theorem:

Diagonals bisect each other at right angles

Half diagonals: cm and cm

Side² = 12² + 9² = +

Side² =

Side = cm

Excellent! Using Pythagoras: side = 15 cm.

9. In a parallelogram, one angle is 60°. Find all other angles.

Let the angle be A = °

Adjacent angle B = 180 - 60 = °

Opposite angle C = A = °

Opposite angle D = B = °

Perfect! Angles are 60°, 120°, 60°, 120°.

10. Write the relationship between the sides and diagonals of a rectangle.

If length = l and breadth = b

Diagonal² = l² + b²l

This uses Pythagorasangle sum theorem

Both diagonals are equalunequal in length

Excellent! Diagonal = √(l² + b²) by Pythagoras theorem.

Part A: Section C – Long Answer Questions (4 Marks Each)

1. In quadrilateral ABCD, the angles are in ratio 1 : 2 : 3 : 4. Find all angles and name the type of quadrilateral if possible.

Step 1: Find angles

Let angles be x, 2x, 3x, 4x

Sum = x + 2x + 3x + 4x = 10x10

10x =

x =

Angles are:

∠A = °, ∠B = °, ∠C = °, ∠D = °

Step 2: Type

All angles are differentsame

Type: IrregularRegular quadrilateral (no special name)

Perfect! It's an irregular quadrilateral with unequal angles.

2. Prove that diagonals of a rectangle are equal and bisect each other.

Draw rectangle ABCD with diagonals AC and BD

Proof of equality:

In triangles ABC and DCB

AB = DCBC (opposite sides)

BC = BCAB (common side)

∠ABC = ∠DCB = °

By SASASA congruence, triangles are equal

Therefore AC = BDAB (diagonals equal)

Proof of bisection:

Rectangle is a parallelogramtrapezium

Diagonals of parallelogram bisectare equal to each other

Excellent! Proved both properties using congruence and parallelogram properties.

3. Write the differences and similarities between rectangle, square, rhombus, and parallelogram in tabular form.

Create a comparison table:

All sides equal:

Rectangle: NoYes, Square: YesNo, Rhombus: YesNo, Parallelogram: NoYes

All angles 90°:

Rectangle: YesNo, Square: YesNo, Rhombus: NoYes, Parallelogram: NoYes

Diagonals equal:

Rectangle: YesNo, Square: YesNo, Rhombus: NoYes, Parallelogram: NoYes

Diagonals perpendicular:

Rectangle: NoYes, Square: YesNo, Rhombus: YesNo, Parallelogram: NoYes

Similarity: All have opposite sides equalunequal and parallelperpendicular

Perfect! Comprehensive comparison completed.

4. Draw a parallelogram ABCD, and prove that opposite sides are equal.

Draw parallelogram ABCD with diagonal AC

Given: ABCD is a parallelogram

To prove: AB = CDBC and AD = BCAB

Proof: In triangles ABC and CDA

AC = ACBD (common side)

∠BAC = ∠DCA (alternate angles, AB parallelperpendicular to CD)

∠BCA = ∠DAC (alternate angles, BC parallelperpendicular to AD)

By ASASAS congruence, triangles are congruent

Therefore AB = CDBC and BC = ADAB

Excellent! Proved using alternate angles and ASA congruence.

5. The diagonals of a rhombus are 16 cm and 12 cm. Find its side using Pythagoras theorem.

Given: Diagonals d₁ = cm, d₂ = cm

Property: Diagonals bisect each other at °

Half diagonals:

d₁/2 = 16/2 = cm

d₂/2 = 12/2 = cm

Using Pythagoras theorem:

Side² = 8² + 6²

Side² = +

Side² =

Side = cm

Perfect! Using Pythagoras theorem, side = 10 cm.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. Diagonals of a rectangle are:

(a) Unequal (b) Equal (c) Perpendicular (d) Parallel

Correct! Rectangle diagonals are equal in length.

7. A parallelogram with all angles 90° is called:

(a) Square (b) Rectangle (c) Rhombus (d) Kite

Perfect! Rectangle is a parallelogram with all angles 90° (Square needs all sides equal too).

8. A kite has:

(a) Two pairs of adjacent sides equal (b) Opposite sides equal

(c) All sides equal (d) No equal sides

Excellent! Kite has two pairs of consecutive/adjacent sides equal.

9. Diagonals of a square are:

(a) Equal and perpendicular (b) Unequal (c) Equal only (d) Perpendicular only

Perfect! Square diagonals have both properties: equal AND perpendicular.

10. The diagonals of a rectangle bisect each other at:

(a) 60° (b) 45° (c) 90° (d) Not necessarily right angles

Correct! Rectangle diagonals bisect each other but not necessarily at 90° (unless it's a square).

🏆 Outstanding Achievement! You've Mastered Advanced Quadrilaterals!

Here's what you've conquered at the hard level:

• Advanced Definitions:

  Kite:

  • Two pairs of adjacent (consecutive) sides equal
  • One diagonal bisects the other at 90°
  • One pair of opposite angles equal

  Rhombus:

  • Parallelogram with all sides equal
  • Diagonals perpendicular and bisect each other
  • Opposite angles equal

  Relationships:

  • Square = Rectangle + Rhombus properties
  • Rectangle = Parallelogram + all angles 90°
  • Rhombus = Parallelogram + all sides equal

• Proof Techniques:

  Congruence Method:

  • Use SAS, ASA, or SSS congruence
  • Prove triangles equal to show sides/angles equal
  • Example: Proving rectangle diagonals equal

  Property Method:

  • Use known properties of parent figure
  • Example: Rectangle is parallelogram, so diagonals bisect

  Coordinate/Pythagoras Method:

  • Use Pythagoras for side calculations
  • Example: Finding rhombus side from diagonals

• Complex Angle Problems:

  Ratio problems:

  • Angles in ratio ac:d
  • Sum = (a+b+c+d)x = 360°
  • Example: 35:6 → 18x = 360° → x = 20°

  Relationship in parallelogram:

  • Opposite angles equal
  • Adjacent angles supplementary (sum = 180°)
  • All angles sum to 360°

• Diagonal Calculations:

  Rhombus side from diagonals:

  • Diagonals bisect at 90°
  • Use half-diagonals
  • Side² = (d₁/2)² + (d₂/2)²
  • Example: d₁=16, d₂=12 → side = 10 cm

  Rectangle diagonal:

  • Diagonal² = length² + breadth²
  • Both diagonals equal
  • Use Pythagoras theorem

• Hierarchical Relationships: