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# Arithmetic Sequences

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The interior angles of a quadrilateral form an arithmetic sequence. If the measure of the largest angle is 129 degrees, what is the measure of the second largest angle, in degrees?

Thanks :D

#1
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Sum = [First term + last term] / 2 x Number of terms

360  =[F + 129] / 2 x 4, solve for F

720 =[F + 129] x 4

180 =[F + 129]

F = 180 - 129

F =51 degrees - the smallest angle

N =[L - F] / D + 1

4 =[129 - 51] / D

3 = /D

3D = 78

D = 78/3

D =26 - diffrence between the 4 angles

51+26 =77 degrees - the second smallest angle

77+26 =103 degrees - the second largest angle. So that you have:

51, 77, 103, 129 angles in degrees.

Jan 9, 2018
#2
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The interior angles of a quadrilateral  sum to 360°

Let  a1 be the smallest angle   and d  be the common difference between the angles

So   we have the following equation

129  =  a1 + 3d    ⇒  129 - 3d  =  a1     (1)

And

a1 + (a1 + d)  + (a1 + 2d )  +  129   =   360     (2)

Sub  (1)  into (2)  and simplifying we have that

129 - 3d +  (129 - 2d)  + ( 129 - d) +  129   =  360

516  - 6d   =  360      rearrrange as

516  -  360  =  6d

156   =  6d          divide both sides by 6

26   = d

And using  (1) we have that   129 - 3(26)    =  a1     =  51°

So.....the second largest angle is    51  + 2(26)   =  103°   Jan 9, 2018