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The measure of angles of quadrilateral ABCD form the arithmetic sequence ∠A, ∠B, ∠C, ∠D. If the measure of angle B is 72 degrees what is the degree measure of angle D?

 Jun 27, 2021
 #1
avatar+234 
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Because a quadrilateral angles sum up to 360 degrees, A, B, C, and D must sum to 360.

 

Since it's a arithmetic sequence with B being 72, we can write them as 

A, B, C, D
72-x, 72, 72+x, 72+2x

 

Add them all up to get, 288+2x = 360. Simplify to get x = 36, and angle D was equal to 72 + 2x, so angle D is 144 degrees.

 Jun 27, 2021
 #2
avatar+736 
0

Let $\angle{A} = x.$

Let the change in angles be $n.$

We have: $x+(x+n)+(x+2n)+(x+3n) = 360$

$x+n = 72$

We want to find $x+3n.$

Rewrite first equation: $3x+5n + 72 = 360$

$3x + 5n = 288$

Rewrite third equation: $3x + 3n = 216$

$2n = 72$

$3n = \frac{3}{2} \cdot 72 = 108$

$n = 36$

$x + 36 = 72$

$x = 36$

$x + 3n = 36 + 108 = \boxed{144}$

 Jun 27, 2021

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