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.

The word “Quadrilateral” derives from “Quad” & “Lateral” which means “Four Sides”.

Hence, a closed shape with four sides is known as Quadrilateral

(a) four straight sides
(b) four angles
(c) four vertex

## 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

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

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