Systems of Equations but With Angle Measures

Question

"A radio tower is supported by cables staked at the same point in the ground. <(pretend that's an angle sign)BDC and <CBD are complementary angles. <ADC and <CAD are also complementary angles. The angle measures are as follows: m<BDC=(y-2)°, m<CBD=(2x+3)°, m<CAD=(2x-y)°, and m<ADC=(x-1)°. At what angle measures do the cables reach the ground?" Now, I've already redone this problem like 3 times and I'm still getting the same answers, "m<BDC=42.5°, m<CBD=92°, m<CAD=151.66°, and m<ADC=29.33°" but apparently my teacher said that he doesn't think that they should be decimals(he hasn't done it himself yet) and told me to google systems of equations, which I did, but I still don't get it. Can someone explain this to me, please?

Answer 1

<BDC=y-2

<CBD=2x+3

<CAD=2x-y

<ADC=x-1

(y-2)+(2x+3)=90

(2x-y)+(x-1)=90

y+2x-2+3=90

2x+x-y-1=90

y+2x+1=90

-y+3x-1=90

add the equations; the y's will cancel and the 1's will cancel

5x=180

x=180/5

x=36º

using y+2x+1=90...

y+72+1=90

y+73=90

y=90-73

y=17º

now substitute into y-2, 2x+3, 2x-y, and x-1

<BDC=17-2=15º

<CBD=2*36+3=72+3=75º

<CAD=2*36-17=72-17=55º

<ADC=36-1=35º

check: 15+75=90

           55+35=90

Answer 2

This can indeed be solved as system of equations problem.

 

Being complementary angles, they have a sum of 90 so the equation from <BDC and <CBD, the equation is:

 

E1: (y-2) + (2x+3) = 90

 

By the same notion, from <ADC and <CAD, the equation is:

 

E2: (2x-y) + (x-1) = 90

 

E1 simplifies to: 2x +y = 89 and E2 simplifies to 3x - y = 91

 

So, E1 + E2 gives the equation 5x = 180.

 

So, x = 36.

 

You can then solve for y and determine the angles.