**Quadrilaterals**Simple Classification of Quadrilaterals. For detailed version check at wikipedia |

**Q1: Define quadrilateral.**

Answer: A quadrilateral is the union of four line-segments determined by four distinct coplanar points of which no three are collinear and the line-segments intersect only at end points.

**Q2: What are the three conditions essential to construct a planar quadrilateral?**

Answer: The three conditions are:

- The four points A, B, C and D must be distinct and co-planar.
- No three of points A, B, C and D are co-linear.
- Line segments i.e. AB, BC, CD, DA intersect at their end points only.

**Q3: What is a convex quadrilateral?**

Answer: If in a given quadrilateral, no side intersects the line to its opposite side, then the quadrilateral is said to be a convex quadrilateral. The diagonals of a convex quadrilateral intersect each other.

**Q4: If four points are co-linear, can it be a quadrilateral?**

Answer: No. It would be a line segment.

**Q5: What is the angle sum property of quadrilateral?**

Answer: Sum of interior angles of a quadrilateral equals 360°.

Sum of exterior angles of a quadrilateral equals 360°.

As shown in figure above,

sum of interior angles, ∠1 + ∠2 + ∠3 + ∠4 = 360°

and sum of exterior angles, ∠5 + ∠6 + ∠7 + ∠8 = 360°

**Q6: Prove that the sum of interior angles of a quadrilateral equals 360°.**

Answer: Let ABCD be a quadrilateral and AC be a diagonal.

Using angle sum property of a triangle, in Δ ABC,

∠BAC + ∠ACB + ∠B = 180° ...(i)

Similarly in ΔADC,

∠DAC + ∠ACD + ∠D = 180° ...(ii)

Adding (i) and (ii), we get

∠BAC + ∠ACB + ∠B + ∠DAC + ∠ACD + ∠D = 180° + 180°

∠BAC + ∠DAC + ∠B + ∠ACB + ∠ACD + ∠D = 360°

∵ ∠BAC + ∠DAC = ∠A and ∠ACB + ∠ACD = ∠A, we have

∠A + ∠B + ∠C + ∠D = 360°

**Q7(NCERT): The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.**

Answer: Let the common ration be

*p*.

∴ Angles are: 3p, 5p, 9p and 13p.

Using angle sum property of quadrilateral i.e. sum of interior angles of a quadrilateral equals 360°.

⇒ 3p + 5p + 9p + 13p = 360°

⇒ 30p = 360°

⇒ p = 12°

∴ Angles are:

3p = 3 × 12° = 36°

5p = 5 × 12° = 60°

9p = 9 × 12° = 108°

3p = 13 × 12° = 156°

**Q8(CBSE 2011): Angles of a quadrilateral are in the ratio 3 : 6 : 8: 13. The largest angle is :**

(a) 178°

(b) 90°

(c) 156°

(d) 36°

Answer: (c) 156°

3p + 6p + 8p + 13p = 30p = 360° ⇒ p = 12° Largest angle is 13p = 13 × 12° = 156°

**Q9: How will you define opposite sides of a quadrilateral?**

Answer: The opposite sides of a quadrilateral are two of its sides which have no common point.

As shown in Fig-5, AB, CD form pair of opposite sides. Similarly, AD, BC form another pair of opposite sides.

**Q10: What are the adjacent or consecutive sides of a quadrilateral?**

Answer: The consecutive sides of a quadrilateral are two of its sides that have a

__common end point__.

As shown in Fig-5, AD, AB for one pair of consecutive sides.

The other three pairs of consecutive sides are: (AB, BC), (BC,CD) and (CD, DA)

**Q11: Identify the opposite angles in quadrilateral shown in Fig-5.**

Answer: The opposite angles of a quadrilateral are two of its angles which do not include a side in their intersection.

In Fig-5, ∠A, ∠C form one pair of opposite angles. ∠B, ∠D form another pair of opposite angles.

**Q12: Identify the consecutive angles in quadrilateral shown in Fig-5.**

Answer: Pair of consecutive angles are: (∠A, ∠B), (∠B, ∠C), (∠C, ∠D) and (∠D, ∠A).

**Q13: Three angles of a quadrilateral are 75º, 90º and 85º. The fourth angle is**

(a) 90º

(b) 85º

(c) 105º

(d) 110º

Answer: (d) 110º

**Q14: Define Parallelogram.**

Its properties are:

- The opposite sides of a parallelogram are equal.
- The opposite angles of a parallelogram are equal.
- The diagonals of a parallelogram bisect each other.

**Q15: What is a Trapezium?**

Answer: A quadrilateral which has one pair of opposite sides parallel is called a trapezium.

**Q16: What is an isosceles trapezium?**

rhombus |

**Q17: What is a rhombus?**

Answer: A rhombus is a parallelogram in which any pair of adjacent sides is equal.

Properties of a rhombus:

- All sides of a rhombus are equal
- The opposite angles of a rhombus are equal
- The diagonals of a rhombus bisect each other at right angles.

rectangle |

**Q18: What is a rectangle?**

Answer: A parallelogram which has one of its angles a right angle is called a rectangle.

Properties of a rectangle are:

- The opposite sides of a rectangle are equal
- Each angle of a rectangle is a right-angle.
- The diagonals of a rectangle are equal.
- The diagonals of a rectangle bisect each other.

**Q19: What is a square?**

Answer: A square is a rectangle, with a pair of adjacent sides equal.

Or

A quadrilateral all of whose sides are equal and all of whose angles are right angles.
Or

A a parallelogram having all sides equal and each angle equal to a right angle is called a square.square |

Properties of square are:

- All the sides of a square are equal.
- Each of the angles measures 90°.
- The diagonals of a square bisect each other at right angles.
- The diagonals of a square are equal.

**Q20(CBSE 2011): All the angles of a convex quadrilateral are congruent. However, not all its sides are congruent. What type of quadrilateral is it?**

(a) parallelogram

(b) square

(c) rectangle

(d) trapezium

Answer: (c) rectangle

(In progress...)

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