What is a Quadrilateral?
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AB, BC, CD and DA are the four sides having ∠ A, ∠ B, ∠ C and ∠ D as four angles while AC is the diagonal of the quadrilateral ABCD. A quadrilateral is a polygon that has four vertices and four sides enclosing 4 angles and the sum of all the angles is 360°.
Quadrilaterals
When we draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180o. Therefore the total angle sum of the quadrilateral is 360o.
Read More:NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals
Types of Quadrilateral
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The Quadrilaterals can be classified into different types as follows.
Rectangle
The rectangle is a quadrilateral where the opposite sides are equal and the angle between the adjacent sides is a right angle.
Rectangle
Square
The Square is a quadrilateral where all the sides are equal and the angle between the adjacent sides is a right angle.
Square
Parallelogram
The Parallelogram is a quadrilateral where the opposite sides are parallel and the opposite angles are congruent. The Diagonals of a parallelogram bisect each other.
Parallelogram
Rhombus
The Rhombus is a quadrilateral where the opposite sides are parallel and all sides are equal. The Diagonals of a rhombus bisect each other at right angles. It is also called as a kite.
Rhombus
Trapezium
The Trapezium is a quadrilateral where one pair of the opposite sides are parallel and the length of the diagonals are equal.
Trapezium
Read More:Perimeter of a parallelogram
Quadrilateral Angle Sum Property Theorem
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According to the property, the sum of all four interior angles is 360o.
Proof: In the quadrilateral ABCD
- ∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles.
- AC is a diagonal.
- AC divides the quadrilateral into two triangles, ΔABC and ΔADC
Quadrilateral
Consider the triangle ADC
∠D + ∠DAC + ∠DCA = 180°…(i) ( Sum of angles in a triangle)
Now consider triangle ABC,
∠B + ∠BAC + ∠BCA = 180° …(ii) ( Sum of angles in a triangle)
On adding both the equations (i) and (ii) we have
(∠D + ∠DAC + ∠DCA) + (∠B + ∠BAC + ∠BCA) = 180° + 180°
∠D + (∠DAC + ∠BAC) + (∠BCA + ∠DCA) + ∠B = 360°
Since
(∠DAC + ∠BAC) = ∠DAB and
(∠BCA + ∠DCA) = ∠BCD
On substitution, we have
∠D + ∠DAB + ∠BCD + ∠B = 360°
Here,
∠DAB = ∠A
∠BCD = ∠C
Thus we have,
∠D + ∠A + ∠C + ∠B = 360°
Thus the sum of angles of a quadrilateral is 360°
Also Read:
Things to Remember
- According to the Quadrilateral angle sum property theorem, the total sum of the interior angles of a quadrilateral is 360°.
- A quadrilateral is formed by joining four non-collinear points.
- A quadrilateral has four sides, four vertices and four angles.
- Rectangle, Square, Parallelogram, Rhombus, Trapezium are some of the types of quadrilaterals.
- In a Square and Rectangle, the diagonals intersect at right angles.
- In a parallelogram and rhombus, the opposite sides are parallel.
- In a trapezium, only one pair of opposite sides are parallel.
Sample Questions
Ques. Find the fourth angle of a quadrilateral whose angles are 90 degrees, 45 degrees, and 60 degrees. (2 marks)
Ans. The sum of all the interior angles of a quadrilateral is 360 degrees
Let the unknown angle be x
We have,
90o + 45o + 60o + x = 360o
195o + x = 360o
x = 165o
Ques. If the sum of three interior angles of a quadrilateral is 240o, find the fourth angle. (2 marks)
Ans. Let us assume the fourth angle as x since the sum of four interior angles of a quadrilateral is 360o
Thus , x + 240o = 360o
x = 120o
Ques. If the angles of a quadrilateral are in the ratio 6 : 3: 4:5, determine the value of the four angles. (2 marks)
Ans. Let the angles be 6x, 3x, 4x and 5x
According to the angle sum property of the quadrilateral
6x + 3x + 4x + 5x = 360o
18x = 360o
x = 20o
Thus the four angles are 120o, 60o, 80o and 100o.
Ques. In a quadrilateral, if the sum of two angles is 200o, find the measure of the other two equal angles. (2 marks)
Ans. Let us say the measure of equal angles is x
Since the sum of interior angles of a quadrilateral is 360o
Thus x +x + 200o= 360o
→ 2x + 200o = 360o
x = 80o
Thus the measure of each angle is 80o
Ques. Explain the angle sum property of a quadrilateral. (1 mark)
Ans. The angle sum property of a quadrilateral states that the sum of all four interior angles of a quadrilateral is 360o
Ques. What are the applications of the angle sum property of a triangle? (1 mark)
Ans. We can use this concept in other geometric proofs, such as the sum of all the interior angles of a quadrilateral.
Ques. What is the formula of the sum of the interior angles of a polygon with n number of sides? (1 mark)
Ans. The formula of the sum of the interior angle = (n -2) * 180o
Ques. What is the measure of the sum of the exterior angles of a quadrilateral? (2 marks)
Ans. We know that the sum of all the exterior angles of a polygon is 360o. A quadrilateral is a polygon with four sides. Therefore, the sum of the exterior angles of a quadrilateral is 360o.
Ques. If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. Can you reason out why? (2 marks)
Ans.Let sides AB and CD of the quadrilateral ABCD be equal and also AD=BC. Draw diagonal AC
Δ ABC ≅ Δ CDA
so, ∠ BAC = ∠ DCA
and ∠ BCA = ∠ DAC
ABCD is a parallelogram, where each pair of opposite sides is equal and conversely if each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
Ques. If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram. Explain. (2 marks)
Ans. Here both its diagonals are intersecting at point O . Measure the length of OA, OB, OC and OD
you will observe that OA = OC and OB = OD
Thus O is the midpoint of both the diagonals. Repeat this activity with some more parallelograms, each time you will find that O is the midpoint of both the diagonals.
Ques. If the diagonals of a parallelogram are equal, then show that it is a rectangle. (2 marks)
Ans.Let ABCD is a parallelogram such that AC = BD.
In ΔABC and ΔDCB,
AC = DB [Given]
AB = DC [Opposite sides of a parallelogram]
BC = CB [Common]
∴ ΔABC ≅ ΔDCB [By SSS congruency]
⇒ ∠ABC = ∠DCB [By C.P.C.T.] …(1)
Now, AB || DC and BC is a transversal. [ Δ ABCD is a parallelogram]
∴ ∠ABC + ∠DCB = 180° … (2) [Co-interior angles]
From (1) and (2), we have
∠ABC = ∠DCB = 90°
i.e., ABCD is a parallelogram having an angle equal to 90°.
∴ ABCD is a rectangle.
Ques. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus. (2 marks)
Ans. Let ABCD be a quadrilateral such that the diagonals AC and BD bisect each other at right angles at O.
∴ In ΔAOB and ΔAOD, we have
AO = AO [Common]
OB = OD [O is the mid-point of BD]
∠AOB = ∠AOD [Each 90]
∴ ΔAQB ≅ ΔAOD [By,SAS congruency
∴ AB = AD [By C.P.C.T.] ……..(1)
Similarly, AB = BC .. .(2)
BC = CD …..(3)
CD = DA ……(4)
∴ From (1), (2), (3) and (4), we have
AB = BC = CD = DA
Thus, the quadrilateral ABCD is a rhombus.
Ques. If an angle of a parallelogram is two-third of its adjacent angle, then find the smallest angle of the parallelogram. (2 marks)
Ans. In a parallelogram ABCD,
Let∠Abe x and ∠B be\(\frac{2}{3}\)x
∴∠A +∠B = 180o
⇒ x +\(\frac{2}{3}\)x =180o
⇒\(\frac{5}{3}\)x =180o
⇒ xo=180o*\(\frac{3}{5}\) = 108o
∠A =108o,∠A =\(\frac{2}{3}\)* 108o= 72o
Ques. PQRS is a parallelogram, in which PQ = 12 cm and its perimeter is 40 cm. Find the length of each side of the parallelogram. (2 marks)
Ans. Here, PQ = SR = 12 cm
Let PS = x and PS =QR
Therefore,
x + 12 + x +12 = perimeter
2x + 24 = 40
2x = 16
x = 8
Hence the length of each side of the parallelogram is 12 cm, 8cm, 12 cm and 8 cm.
Ques. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see figure). Show that (2 marks)
Ans.
(i) In ΔAPB and ΔCQD, we have
∠APB = ∠CQD [Each 90°]
AB = CD [ Δ Opposite sides of a parallelogram ABCD are equal]
∠ABP = ∠CDQ
[ Δ Alternate angles are equal as AB || CD and BD is a transversal]
∴ ΔAPB = ΔCQD [By AAS congruency]
(ii) Since, ΔAPB ≅ ΔCQD [Proved]
⇒ AP = CQ [By C.P.C.T.]
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