Understanding Quadrilaterals

Quadrilaterals are closed figures with four sides, four angles, and four vertices.

Key Concepts:

• Types of Quadrilaterals:
  • Trapezium: A quadrilateral with at least one pair of parallel sides.
  • Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
  • Rectangle: A parallelogram with all angles right angles.
  • Square: A rectangle with all sides equal.
  • Rhombus: A parallelogram with all sides equal.
  • Kite: A quadrilateral with two pairs of adjacent sides equal.
• Properties of Quadrilaterals:
  • Sum of interior angles is 360 degrees.
  • Diagonals: Line segments joining opposite vertices.
  • Properties of different types of quadrilaterals (e.g., opposite sides equal, diagonals bisect each other, etc.)
• Special Properties:
  • Relationship between sides, angles, and diagonals in different quadrilaterals.
  • Area and perimeter formulas for specific quadrilaterals.

Visualizing and understanding the properties of different quadrilaterals is crucial for solving problems related to geometry and mensuration.

Exercise 3.1

1. Given here are some figures.

 Classify each of them on the basis of the following.

 (a) Simple curve (b) Simple closed curve (c) Polygon (d) Convex polygon (e) Concave polygon

Ans : 

Figures (1), (2), (5), and (6) are simple curves.

Figures (1), (2), (5), and (6) are simple closed curves.

Figures (1) and (2) are polygons.

Figure (2) is a convex polygon.

Figures (1) and (4) are concave polygons.

2.  What is a regular polygon?

 State the name of a regular polygon of 

(i) 3 sides (ii) 4 sides (iii) 6 sides

Ans : 

Names of Regular Polygons:

• (i) 3 sides: Equilateral triangle
• (ii) 4 sides: Square
• (iii) 6 sides: Regular hexagon

Exercise 3.2

1. Find x in the following figures.

Ans : 

Figure (a)

In this figure, we have two angles given: 125° and 125°. We know that the sum of the exterior angles of any polygon is 360°.

So, 125° + 125° + x = 360° 

250° + x = 360°

 x = 360° – 250° 

x = 110°

Figure (b)

In this figure, we have three angles given: 60°, 90°, and 70°. There is a right angle indicated, which is 90°. 

So, 60° + 90° + 70° + x + 90° = 360° 

310° + x = 360° 

x = 360° – 310°

 x = 50°

2. Find the measure of each exterior angle of a regular polygon of

(i) 9 sides

(ii) 15 sides

Ans : 

(i) 9 sides

Measure of each exterior angle = 360°/9 = 40°

(ii) 15 sides

Measure of each exterior angle = 360°/15 = 24°

3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Ans : 

We know that the sum of the exterior angles of any polygon is 360°.

Let the number of sides be n.

So, n * 24° = 360°

To find n, divide both sides by 24°: n = 360° / 24° n = 15

Therefore, the regular polygon has 15 sides.

4. How many sides does a regular polygon have if each of its interior angles is 165°?

Ans : 

Solution:

• Given: Interior angle = 165°
• To find: Number of sides

Approach:

1. Find the exterior angle.
2. Use the formula: Number of sides = 360°/exterior angle

• Step 1: Find the exterior angle
  • The sum of the interior angle and its corresponding exterior angle is 180°.
  • Exterior angle = 180° – Interior angle 
  • = 180° – 165° = 15°
• Step 2: Find the number of sides
  • Number of sides = 360° / Exterior angle = 360° / 15° = 24

Therefore, the regular polygon has 24 sides.

5. (a) Is it possible to have a regular polygon with measure of each exterior angle a is 22°?

(b) Can it be an interior angle of a regular polygon? Why?

Ans :

a) 

No, it is not possible.

• To find the number of sides, we divide 360 by the exterior angle measure.
• In this case, 360 / 22 is not a whole number.

Since the number of sides must be a whole number, it’s impossible to have a regular polygon with an exterior angle of 22°.

(b)

No, it cannot be an interior angle of a regular polygon.

• The minimum interior angle of a regular polygon is 60° (in an equilateral triangle).
• An interior angle of 22° is less than the minimum possible value.

In conclusion, neither 22° as an exterior angle nor as an interior angle is possible for a regular polygon.

6. (a) What is the minimum interior angle possible for a regular polygon? Why?

(b) What is the maximum exterior angle possible for a regular polygon?

Ans : 

(a)

The minimum interior angle possible for a regular polygon is 60 degrees.

This occurs in an equilateral triangle, which is a regular polygon with three sides. 

As the number of sides in a regular polygon increases, the interior angle also increases. So, 60 degrees is the smallest possible interior angle for a regular polygon.

(b) 

This occurs in an equilateral triangle. The exterior angle of an equilateral triangle is 180° – 60° = 120°.

As the number of sides in a regular polygon increases, the exterior angle decreases. So, 120 degrees is the largest possible exterior angle for a regular polygon.

Remember: The sum of an interior angle and its corresponding exterior angle is always 180 degrees.

Exercise 3.3

1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.

(i) AD = …………

(ii) ∠DCB = ………

(iii) OC = ………

(iv) m∠DAB + m∠CDA = ……..

Ans : 

Certainly, let’s complete the statements about parallelogram ABCD:

(i) AD = BC Opposite sides of a parallelogram are equal

(ii) ∠DCB = ∠DAB Opposite angles of a parallelogram are equal

(iii) OC = OA Diagonals of a parallelogram bisect each other

(iv) m∠DAB + m∠CDA = 180°

2. Consider the following parallelograms. Find the values of the unknowns x, y, z.

Ans : 

Let’s find the values of x, y, and z in the given parallelograms:

Parallelogram (i)

• y = 100° (Opposite angles of a parallelogram are equal)
• x + 100° = 180° (Adjacent angles of a parallelogram are supplementary)
  • x = 80°
• z = x = 80° (Opposite angles of a parallelogram are equal)

Parallelogram (ii)

• y = 50° (Alternate interior angles)
• x + y + 50° = 180° (Angle sum property of a triangle)
  • x + 100° = 180°
  • x = 80°
• z = x = 80° (Alternate interior angles)

Parallelogram (iii)

• x = 90° (Vertically opposite angles)
• x + y + 30° = 180° (Angle sum property of a triangle)
  • 90° + y + 30° = 180°
  • y = 60°
• z = y = 60° (Alternate interior angles)

Parallelogram (iv)

• x = 100° (Opposite angles of a parallelogram are equal)
• y = 80° (Alternate interior angles)
• z = 80° (Alternate interior angles)

3. Can a quadrilateral ABCD be a parallelogram if

(i) ∠D + ∠B = 180°?

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

(iii) ∠A = 70° and ∠C = 65°?

Ans : 

Analyzing Quadrilaterals

(i) 

No, a quadrilateral cannot be a parallelogram if only the sum of two opposite angles is 180°. A parallelogram has both pairs of opposite angles equal, and the sum of adjacent angles is 180°.

(ii) 

No, a quadrilateral cannot be a parallelogram if the opposite sides are not equal. A parallelogram has opposite sides equal in length.

(iii)

No, a quadrilateral cannot be a parallelogram if the opposite angles are not equal. A parallelogram has opposite angles equal.

In conclusion, none of the given conditions are sufficient to determine that a quadrilateral is a parallelogram.

4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

Ans : 

5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

Ans : 

Let the two adjacent angles be 3x and 2x.

Sum of adjacent angles of a parallelogram is 180°.

 So, 3x + 2x = 180° 

Combine like terms: 5x = 180° 

Divide both sides by 5: x = 36°

Therefore, one angle is 3x = 3 * 36° = 108° 

And the other angle is 2x = 2 * 36° = 72°

Since opposite angles of a parallelogram are equal, the angles of the parallelogram are 108°, 72°, 108°, and 72°.

6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Ans : 

Let’s denote the measure of each of the equal adjacent angles as x.

Since the sum of adjacent angles of a parallelogram is 180°, we can write the equation:

 x + x = 180° 

2x = 180° x

 = 90°

Therefore, each of the adjacent angles measures 90°.

Since opposite angles of a parallelogram are equal, all angles of the parallelogram are 90°.

Hence, the parallelogram is a rectangle, and all its angles measure 90°.

7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

Ans :
Analysis:

1. Angle z:
   
   • We know that the angle EHP is 40 degrees.
   • Since opposite angles in a parallelogram are equal, angle HOP is also 40 degrees.
   • Angle z is supplementary to angle HOP (they form a linear pair).
   • Therefore, z = 180 – 40 = 140 degrees.
2. Angle y:
   
   • We know that the angle POX is 70 degrees.
   • Since opposite angles in a parallelogram are equal, angle HEQ is also 70 degrees.
   • Angle y is supplementary to angle HEQ.
   • Therefore, y = 180 – 70 = 110 degrees.
3. Angle x:
   
   • We know that the angle EHO is 140 degrees (calculated above).
   • Since the sum of adjacent angles in a parallelogram is 180 degrees, angle x + angle EHO = 180 degrees.
   • Therefore, x = 180 – 140 = 40 degrees.

8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)

Ans : 

Parallelogram GUNS

• In a parallelogram, opposite sides are equal.
• Therefore, GU = SN and GS = UN.

So, we have:

• 3x = 18
• 3y – 1 = 26

Solving these equations:

• x = 18 / 3 = 6
• 3y = 26 + 1 = 27
  • y = 27 / 3 = 9

Hence, in parallelogram GUNS, x = 6 cm and y = 9 cm.

Parallelogram RUNS

• In a parallelogram, diagonals bisect each other.
• Therefore, OR = ON and OU = OS.

So, we have:

• x + y = 16
• y + 7 = 20

Solving these equations:

• From the second equation, y = 20 – 7 = 13
• Substituting y in the first equation, x + 13 = 16
  • x = 16 – 13 = 3

Hence, in parallelogram RUNS, x = 3 cm and y = 13 cm.

Therefore, the values of x and y in the given parallelograms are:

• GUNS: x = 6 cm, y = 9 cm
• RUNS: x = 3 cm, y = 13 cm

9. 

In the above figure both RISK and CLUE are parallelograms. Find the value of x.

Ans : 

∠1 = ∠L = 70° (Opposite angles of a parallelogram)

∠K + ∠2 = 180°

Sum of adjacent angles is 180°

120° + ∠2 = 180°

∠2 = 180° – 120° = 60°

In ∆OES,

∠x + ∠1 + ∠2 = 180° (Angle sum property)

⇒ ∠x + 70° + 60° = 180°

⇒ ∠x + 130° = 180°

⇒ ∠x = 180° – 130° 

= 50°

Hence x = 50°

10. Explain how this figure is a trapezium. Which of its two sides are parallel?

Ans : 

∠M + ∠L = 100° + 80° = 180°

∠M and ∠L are the adjacent angles

sum of adjacent interior angles is 180°

KL is parallel to NM

Hence KLMN is a trapezium.

11. Find m∠C in below figure if AB || DC

Ans :
Interior angles on the same side of a transversal are supplementary (their sum is 180°).

Therefore, m∠B + m∠C = 180° 

Substitute the given value of m∠B: 

120° + m∠C = 180° 

Subtract 120° from both sides: 

m∠C = 180° – 120° m∠C = 60°

Angle C is 60 degrees.

12. Find the measure of ∠P and ∠S if SP RQ  in Fig 3.28.

 (If you find m∠R, is there more than one method to find m∠P?)

Ans :
Given:

• Quadrilateral PQRS
• Angle Q = 130°
• SP || RQ (SP is parallel to RQ)

To find:

• Measure of angle P (∠P)
• Measure of angle S (∠S)

Solution:

1. Identifying the shape:

• Since SP is parallel to RQ, and opposite sides of a parallelogram are parallel, the quadrilateral PQRS is a parallelogram.

2. Finding angle P:

• In a parallelogram, opposite angles are equal.
• Therefore, ∠P = ∠Q = 130°

3. Finding angle S:

• The sum of adjacent angles is 180°.
• So, ∠Q + ∠S = 180°
• Substituting the value of ∠Q: 130° + ∠S = 180°
• Therefore, ∠S = 180° – 130° = 50°

Conclusion:

• ∠P = 130°
• ∠S = 50°

Exercise 3.4

1. State whether True or False.

(a) All rectangles are squares.

(b) All rhombuses are parallelograms.

(c) All squares are rhombuses and also rectangles.

(d) All squares are not parallelograms.

(e) All kites are rhombuses.

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(h) All squares are trapeziums.

Ans : 

(a) False

(b) True

(c) True

(d) False

(e) False

(f) True

(g) True

(h) True

2. Identify all the quadrilaterals that have

(a) four sides of equal length

(b) four right angles

Ans : 

a) Four sides of equal length

• Rhombus
• Square

These quadrilaterals have all four sides equal in length.

(b) Four right angles

• Rectangle
• Square

These quadrilaterals have all four angles as right angles (90 degrees).

3. Explain how a square is

(i) a quadrilateral

(ii) a parallelogram

(iii) a rhombus

(iv) a rectangle

Ans : 

(i) A square is a quadrilateral

• A square, by definition, has four sides.
• Therefore, a square is a quadrilateral.

(ii) A square is a parallelogram

• A square has two pairs of opposite sides, and these pairs are parallel.
• Hence, a square is a parallelogram.

(iii) A square is a rhombus

• A square, by definition, has all sides equal.
• Therefore, a square is a rhombus.

(iv) A square is a rectangle

• A square has all angles equal to 90 degrees (right angles).
• Hence, a square is a rectangle.

4. Name the quadrilaterals whose diagonals

(i) bisect each other

(ii) are perpendicular bisectors of each other

(iii) are equal

Ans : 

(i) 

• Parallelogram
• Rectangle
• Rhombus
• Square

(ii) 

• Rhombus
• Square

(iii) 

• Rectangle
• Square

5. Explain why a rectangle is a convex quadrilateral.

Ans : 

All interior angles of a rectangle are less than 180 degrees. Specifically, each angle is 90 degrees. For a quadrilateral to be convex, all interior angles must be less than 180 degrees.

The diagonals of a rectangle lie entirely within the quadrilateral. This is a characteristic of convex shapes.

6. ABC is a right-angled triangle and O is the midpoint of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).

Ans : 

1. Constructing the Rectangle:
   • Draw lines parallel to AB through point O and parallel to BC through point O. These lines intersect at point D, forming a rectangle ABCD.
2. Properties of Rectangles:
   • Point O is the midpoint of diagonal AC, which means OA = OC.
   • Point O is also the midpoint of diagonal BD, which means OB = OD.
3. Equidistance:
   • Since OA = OC and OB = OD, and the diagonals of a rectangle are equal in length (AC = BD), we can conclude that OA = OB = OC.

Therefore, point O is equidistant from A, B, and C.