|Simple Classification of Quadrilaterals.|
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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 :
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.
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
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.
Answer: A quadrilateral which has one pair of opposite sides parallel is called a trapezium.
Q16: What is an isosceles trapezium?
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.
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.
OrA quadrilateral all of whose sides are equal and all of whose angles are right angles.
OrA a parallelogram having all sides equal and each angle equal to a right angle is called a 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.
Answer: (c) rectangle