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The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is five
times the measure of the first angle, The third angle is 14 more than the second, Let x, y, and z represent the measures of
the first, second, and third angles, respectively. Find the measures of the three angles.

Sagot :

Cinderofsoulsssidn

Answer:

x = 30

y = 68

z = 82

Step-by-step explanation:

Let the first angle be X, the second angle be Y, and the third angle be Z.

You can write these 3 equations to represent the relationships between the angles:

[tex]x+y+z=180\\y+z=5x\\z=y+14[/tex]

Step 1. Z is already isolated in the 3rd equation, so the first thing I would do is use that to eliminate the Z in the 1st and 2nd equations.

[tex]z=y+14\\\\x+y+z=180\\x+y+(y+14)=180\\\\y+z=5x\\y+(y+14)=5x[/tex]

Step 2. Now, simplify both of those and treat them like a system of 2 equations. I'll simplify the top equation first and solve for x:

[tex]x+y+(y+14)=180\\x+y+y+14=180\\x+2y+14=180\\x+2y=166\\x=166-2y[/tex]

Plug that into the other equation:

[tex]y+(y+14)=5x\\y+y+14=5(166-2y)\\2y+14=830-10y\\12y+14=830\\12y=816\\y=68[/tex]

Step 3. Now we have the value of one variable. Use that to solve for x in the previous equation:

[tex]x=166-2y\\x=166-2(68)\\x=166-136\\x=30[/tex]

Step 4. Now we have two variables. Pick any of the original 3 equations you want to solve for z:

[tex]x+y+z=180\\30+68+z=180\\98+z=180\\z=82[/tex]

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Finally, you can confirm that by plugging these into all of the original equations to make sure they meet the conditions.

[tex]30+68+82=180\\180=180\\\\68+82=5(30)\\150=150\\\\82=68+14\\82=82[/tex]

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