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**