Question

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 nine times the measure of the first angle. The third angle is 26 more than the second let x,y, and z represent the measures of the first second and third angles, find the measures of the three angles

Answer 1

• x = 18, y = 68, z = 94.

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Set equations as per given details.

The sum of the measures of the angles of a triangle is 180:

• x + y + z = 180 (1)

The sum of the measures of the second and third angles is nine times the measure of the first angle:

• y + z = 9x (2)

The third angle is 26 more than the second:

• z = y + 26 (3)

Solution

Substitute the second equation into first:

• x + y + z = 180,
• y + z = 9x.

Solve for x:

• x + 9x = 180,
• 10x = 180,
• x = 18.

Substitute the value of x into second and solve for y:

• y + z = 9x,
• y + z = 9*18,
• y + z = 162,
• y = 162 - z.

Solve the third equation for y:

• z = y + 26,
• y = z - 26.

Compare the last two equations and find the value of z:

• 162 - z = z - 26,
• z + z = 162 + 26,
• 2z = 188,
• z = 94.

Find the value of y:

• y = 94 - 26,
• y = 68.

Answer 2

x = 18°

y = 68°

z = 94°

Explanation:

Define the variables:

• Let x represent the first angle.
• Let y represent the second angle.
• Let z represent the third angle.

Given information:

• 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 nine times the measure of the first angle.
• The third angle is 26 more than the second.

Create three equations from the given information:

`\begin{cases}x+y+z=180\\\;\;\;\;\;\:\: y+z=9x\\\;\;\;\;\;\;\;\;\;\;\;\;\: z=26+y\end{cases}`

Substitute the third equation into the second equation and solve for x:

`\implies y+(26+y)=9x`

`\implies 2y+26=9x`

`\implies x=(2y+26)/(9)`

Substitute the expression for x and the third equation into the first equation and solve for y:

`\implies (2y+26)/(9)+y+26+y=180`

`\implies (2y+26)/(9)+2y=154`

`\implies (2y+26)/(9)+(18y)/(9)=154`

`\implies (2y+26+18y)/(9)=154`

`\implies (20y+26)/(9)=154`

`\implies 20y+26=1386`

`\implies 20y=1360`

`\implies y=68`

Substitute the found value of y into the third equation and solve for z:

`\implies z=26+68`

`\implies z=94`

Substitute the found values of y and z into the first equation and solve for x:

`\implies x+68+94=180`

`\implies x=18`