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.

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