**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:**

- Find the exterior angle.
- 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:**

**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
**.**

**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.

**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 : **

**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.

**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.

**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.**