Math Geometry Question Help?

/_1 and /_2 are supplements.. which means that

/_1 + /_2 = 180

=> (n+15) + (9p) = 180

=> n + 9p = 165 ----------- (i)

since /_2 and /_3 are vertical angles, their measures are equal..

so..

/_2 = /_3

=> 9p = 4n + p

=> 8p = 4n

=> n = 2p

put this value of n in equation (i), to get

2p + 9p = 165

=> 11p = 165

=> p = 165/11

=> p = 15

and n = 2p = 30

now you know n and p, and can find the 3 angles using the relations

Angle 1 = n + 15

Angle 2 = 9p

Angle 3 = 4n + p

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Angle 1 is the supplement of Angle 2 ===> they are supplementary angles.

The sum of the measures of supplementary angles is 180 degrees. Therefore,

( n + 15) + 9p = 180 ===>

n + 15 + 9p = 180 (Equation 1

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Angle 2 and Angle 3 are vertical angles. Vertical angles have the same measures.

Therefore,

9p = 4n + p Equation 2

9p - p = 4n +p - p

8p = 4n

8p/8 = 4n/8

p = 1/2n Equation 3

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Subtitute Equation 3 to Equation 1

n + 15 + 9p = 180 (Equation 1)

n + 15 + 9( 1/2n) = 180

n + 15 + 9/2n = 180

11/2n = 180 - 15

11/2 n = 165

n = 2/11 (165)

n = 330/11

n = 30 degrees ===> value of n

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Substitute the value of n into Equation 2

9p = 4n + p

9p = 4 (30) + p

9p = 120 + p

9p - p = 120 + p - p

8p = 120

p = 120/8

p = 15 degrees ===> value of p

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n = 30 , p = 15

Angle 1 = n + 15 = 30 + 15 = 45 degrees

Angle 2 = 9p = 9(15) = 135 degrees

Angle 3 = 4n + p = 4(30) + 15 = 120 + 15 = 135 degrees

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Supplementary Angles

Angle 1 + Angle 2 = 180

45 + 135 = 180

180 = 180

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Vertical ANgles

Angle 2 = ANgle 3

135 = 135

If angle 2 and angle 3 are vertical, then they are congruent (measures equal each other).

If angle 1 and angle 2 are supplementary, then they add to equal 180 degrees.

Therefore we can create two different equations.

9p = 4n + p (using the two vertical angles)

n + 15 + 9p = 180 (using the supplementary angles)

This is now a system of equations. It appears that substitution is the easiest way to solve. Get n alone in the second equation.

n + 15 + 9p = 180

n = 180 - 15 - 9p

n = 165 - 9p

Now plug in 165 - 9p for n in the first equation.

9p = 4n + p

Becomes

9p = 4(165 - 9p) + p

9p = 660 - 36p + p

9p = 660 - 35p

44p = 660

p = 15

To find n, plug 15 in for p in the equation n = 165 - 9p.

n = 165 - 9(15)

n = 165 - 135

n = 30

To find the measures of the angles plug in the appropriate numbers.

Angle 1 = n + 15 = 30 + 15 = 45

Angle 2 = 9p = 9(15) = 135

Angle 3 = 4n + p = 4(30) + 15 = 120 + 15 = 135

1 is the supplement of angle 2 hence n+15 + 9p = 180

angle 2 & angle 3 are vertical angles hence 9p = 4n + p or n = 2p whence 2p + 15 + 9p = 180 or p = 15 and n = 2(15) = 30

angles are 30+15, 9(15), 4(30) + 15 or 45, 135, 135.

A1 = n +15 ,A 2 = 9p, A3 = 4n +p

A1 = 180 - A2,

A2 = A3 = 9p = 4n + p = 8p = 4n = 2p = n

A1 = 180 - A2 = n + 15 = 180 - 9p = 2p + 15 = 180 - 15

= 2p + 9p = 11p = 165 ------> divided both side by 11

p = 15.

n = 2p = 2*15 = 30

A1 = n +15 = 30 +15 = 45

A2 = 9p = 9 *15 = 135

A3 = 4n + p = 4*30 + 15 = 135