In this chapter we will learn angle sum property of quadrilateral with full derivation.
It is important property of quadrilateral and you have to remember the concept for your exams.
What is Quadrilateral?
The word “Quadrilateral” derives from “Quad” & “Lateral” which means “Four Sides”.
Hence, a closed shape with four sides is known as Quadrilateral
Key Features of Quadrilateral
(a) four straight sides
(b) four angles
(c) four vertex
Examples of Quadrilateral
Angle sum property of Quadrilateral
The property says that the sum of all internal angles of quadrilateral is 360 degree
Given above is the quadrilateral ABCD with four internal angles (∠1, ∠2, ∠3 and ∠4)
According to angle sum property of quadrilateral
∠1 + ∠2 + ∠3 + ∠4 = 360 degree
Proof of Angle Property of Quadrilateral
Given
Quadrilateral ABCD with diagonal BD
To Prove
Sum of Internal angles of Quadrilateral is 180 degree
Solution
Observe that the diagonal BD divides the quadrilateral into two triangles (▵ ABD & ▵BCD)
Taking ▵ABD
We know that sum of angles in triangle adds to 180 degree
∠DAB + ∠ABD + ∠BDA = 180 degree – – – eq. (1)
Similarly taking ▵BCD
∠DBC + ∠BCD + ∠CDB = 180 degree – – – eq. (2)
Adding eq(1) & eq(2), we get;
∠DAB + ∠ABD + ∠BDA + ∠DBC + ∠BCD + ∠CDB = 360 degree
∠DAB + ∠ABC + ∠BCD + ∠CDB = 360 degree
Hence Proved
Practice Questions – Angles of Quadrilateral
(01) Find the angle x in the below quadrilateral
Given
∠ A = 137 degree
∠B = 97 degree
∠ C = 81 degree
To Find
∠D = ?
Solution
We know that sum of angles of quadrilateral is 360 degree
∠ A + ∠B + ∠C + ∠D = 360
Putting the angle values
137 + 97 + 81 + ∠D = 360
∠D = 360 – 315 = 45 degree
Hence, ∠D measures 45 degree
(02) Observe the below quadrilateral and find measure of ∠A
Given:
∠B = 111 degree
∠ C = 90 degree
∠ D = 78 degree
To find
Find the value of ∠ A
Solution
Sum of angles of quadrilateral = 360 degree
∠ A + ∠B + ∠C + ∠D = 360
Putting the values;
∠ A + 111 + 90 + 78 = 360
∠ A = 360 – 279 = 81 degree
Hence ∠ A measures 81 degree
(03) Find the measure of ∠C in the below figure
Given:
∠A = 74 degree
∠B = 23 degree
∠D = 37 degree
To find:
Measure of ∠ C
Solution
The figure ABCD is a quadrilateral with four sides
We know that,
Sum of angles of quadrilateral = 360 degree
∠ A + ∠B + ∠C + ∠D = 360
Putting the values
74 + 23 + x + 37 = 360
x = 360 – 134
x = 226 degree
Hence, ∠C measures 226 degree
(04) Observe the below image and find the measure of ∠ y
Given
∠A = 110 degree
∠B = 125 degree
∠D = 63 degree
To find
Find the measure of angle y
Solution
ABCD is a Quadrilateral.
We know that sum of internal angle of quadrilateral adds to 360 degree
∠ A + ∠B + ∠C + ∠D = 360
Putting the values
110 + 125 + x + 63 = 360
x = 360 – 298 = 62 degree
Hence value of x is 62 degree
Note that DCM is a straight line
We know that angle in a straight lines adds to 180 degree
x + y = 180
Putting the value of angle x
62 + y = 180
y = 180 – 62 = 118 degree
Hence, angle y measures 118 degree
(05) Given below is the quadrilateral ABCD. Find the measure of ∠P, ∠Q, ∠R and ∠S
Given
ABCD is a quadrilateral in which
∠ A = 145 degree
∠ NBC = 81 degree
Ext. ∠ C = 300 degree
To find
∠P, ∠Q, ∠R and ∠S
Solution
(a) Angle measure of ∠P
Note that ABN is a straight line
We know that in straight line, angles add to 180 degree
∠P + 81 = 180
∠P = 180 – 81 = 99 degree
(b) Angle measurement of ∠Q
Given is the exterior angle C = 300
∠Q = 360 – ext.∠C
∠Q = 360 – 300
∠Q = 60 degree
(c) Finding angle measurement of ∠R
We know that ABCD is a quadrilateral.
The sum of angles of quadrilateral is 360 degree
145 + ∠P + ∠Q + ∠R = 360 degree
Putting the values
145 + 99 + 60 + ∠R = 360
∠R = 360 – 304
∠R = 56 degree
(d) Finding ∠S
Note that CDM is a straight line.
We know that angle in a straight line add to 180 degree
∠R + ∠S = 180
56 + ∠S = 180
∠S = 180 – 56
∠S = 124 degree
(06) Given below is quadrilateral ABCD. Find the measure of ∠x, ∠y, ∠z and ∠R
Given
Ext. ∠A = 285 degree
Ext. ∠B = 335 degree
Ext. ∠D = 325 degree
To Find
∠x, ∠y, ∠z and ∠R
Solution
(a) Finding ∠x
∠x = 360 – Ext. ∠A
∠x = 360 – 285 = 75 degree
Hence ∠x measures 75 degree
(b) Finding ∠y
∠y = 360 – Ext. ∠B
∠y = 360 – 335 = 25 degree
Hence ∠y measures 25 degree
(c) Finding ∠z
∠z = 360 – Ext. ∠D
∠z = 360 – 325 = 35 degree
Hence ∠z measures 35 degree
(d) Finding ∠R
we know that sum of angles of quadrilateral adds to 60 degree
∠x + ∠y + ∠z + ∠R = 360
75 + 25 + 35 + ∠R = 360
∠R = 360 -135 = 225
Hence, ∠R measures 225 degree