ANGLES OF A QUADRILATERAL

Like triangles, quadrilaterals have both interior and exterior angles. If we draw a diagonal in a quadrilateral, you divide it into two triangles as shown below. 

Each of the triangle above has interior angles with measures that add up to 180°.

So we can conclude that the sum of the measures of the interior angles of a quadrilateral is 2(180°), or 360°.

Find the values of the variables in the following quadrilaterals :

Example 1 :

Solution :

In a quadrilateral,

Sum of interior angles  =  360

123 + 93 + 72 + x  =  360

x + 288  =  360

x  =  360-288

x  =  72

So, the missing angle is 72.

Example 2 :

Solution :

a + 110  =  180  (linear pair)

a  =  70

<a + 80 + 130 + <b  =  360

70+80+130+<b  =  360

<b  =  360-280

<b  =  80

Example 3 :

Solution :

<a+<a+<a+72  =  360

3<a  =  360-72

3<a  =  288

<a  =  93

So, angle a is 93.

Example 4 :

Solution :

In a quadrilateral,

72+118+90+<a  =  360

280+<a  =  360

<a  =  360-280

<a  =  80

<a+<b  =  180

80+<b  =  180

<b  =  100

Example 5 :

Solution :

<a+100  =  180

<a  =  180-100

<a  =  80

In a quadrilateral,

120+90+<b+80  =  360

290+<b =  360

<b  =  360-290

<b  =  70

<b+<c  =  180

70+<c  =  180

<c  =  110

So, <a  =  80, <b  =  70 and <c  =  110.

Example 6 :

Solution :

In a quadrilateral, sum of interior angles is 360.

2a + a + 90 + (a+16)  =  360

4a+106  =  360

4a  =  360-106

4a  =  244

a  =  61

Example 7 :

Solution :

<a+74  =  180 (linear pair)

<a  =  180-74

<a  =  106

<b+92  =  180(linear pair)

<b  =  180-92

<b  =   88

<a + 74 + <b + <c  =  360

106+74+88+<c  =  360

268+<c  =  360

<c  =  360-268

<c  =  92

<c+<d  =  180

92+<d  =  180

<d  =  180-92

<d  =  88

So, <a  =  106, <b  =  88, <c  =  92 and <d  =  88.

Example 8 :

Two adjacent sides AB and BC of a parallelogram ABCD are in the ratio 5 : 3. If the perimeter is 200 cm, what is the length of AB and BC ?

(a) 25 cm & 50 cm     (b) 40 cm & 37.5 cm

(c) 62.5 cm & 37.5 cm     (d) 60 cm & 62.5 cm

Solution :

The given quadrilateral is in the shape of parallelogram. In the shape parallelogram, the opposite sides will be equal.

The adjacent sides are in the ratio 5 : 3. The sides are 5x and 3x.

Perimeter of parallelogram = 200 cm

2 (5x + 3x) = 200

2(8x) = 200

x = 200/16

x = 12.5

5x = 5(12.5) ==> 62.5 cm

3x = 3(12.5) ==> 37.5 cm

The side lengths are 62.5 cm and 37.5 cm. Option c is correct.

Example 9 :

One angle of a parallelogram is 60°. Find its opposite angle and the adjacent angle. 

Solution :

In a parallelogram, the opposite sides are parallel and equal.

Since the opposite sides are parallel, the sum of co-interior angles is 180 degree.

Let x be the adjacent angle.

60 + x = 180

x = 180 - 60

x = 120

Example 10 :

ABCD is a trapezium with AB || DC. If ∠A = 50°, then find ∠D.

Solution :

∠A = 50°

AB and DC are parallel, then ∠A and ∠D are co-interior angles.

∠A + ∠D = 180

50 + ∠D = 180

∠D = 180 - 50

∠D = 130

Example 11 :

Four angles of quadrilaterals are in the ratio 3 : 4 : 5 : 6. Find its angles.

Solution :

Since the angles are in the ratio 3 : 4 : 5 : 6. Then the angles are 3x, 4x, 5x and 6x.

Sum of the interior angles of a quadrilateral = 360

3x + 4x + 5x + 6x = 360

18x = 360

x = 360/18

x = 20

3x = 3(20) ==> 60

4x = 4(20) ==> 80

5x = 5(20) ==> 100

6x = 6(20) ==> 120

So, the angle measures are 60, 80, 100 and 120.

Example 12 :

ABCD is a rhombus with ∠DAB = 56, determine ∠DBC

Solution :

In rhombus all sides will be equal and adjacent angles are supplementary.

∠DAB + ∠DBC= 180

56 + ∠DBC = 180

∠DBC = 180 - 56

∠DBC = 124

Example 13 :

In the following diagram of isosceles trapezoid ∠A = 46 . Find the measure of all of the remaining angles.

angles-in-quadrilateral-q1

Solution :

∠A = 46

Since the shape is an isosceles trapezoid, two angle measures will be equal.

∠A = 46 and ∠D = 46

∠A + ∠B= 180

∠B = 180-46

∠B = 134

∠C = 134

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