Questions: Find the values of (x, y), and (z) in the figure (in degrees), where (b=2) and (c=14). Step 1 Recall that in a triangle, the sum of the measures of an exterior angle and its adjacent angle is 180,180. and the sum of the measures of the internal angles is (180^circ). Using the above facts and the given figure, write the system of linear equations. [ left beginarrayrlrl x+y+z =180 textEquation 1 x+1.5 z+2 =180 y+1.5 z-14 =180 endarray, quad beginarrayll 180 endarray right. ] Step 2 Rewrite the system in standard form. [ left beginarrayl x+y+z=180 x+1.5 z= y+1.5 z=194 endarray right. ] Adding -1 times the second equation to the first equation produces a new first equation. [ left beginarrayl y-0.5 z= x+1.5 z= y+1.5 z=194 endarray right. ]

Find the values of (x, y), and (z) in the figure (in degrees), where (b=2) and (c=14). Step 1 Recall that in a triangle, the sum of the measures of an exterior angle and its adjacent angle is 180,180. and the sum of the measures of the internal angles is (180^circ). Using the above facts and the given figure, write the system of linear equations. [ left beginarrayrlrl x+y+z =180 textEquation 1 x+1.5 z+2 =180 y+1.5 z-14 =180 endarray, quad beginarrayll 180 endarray right. ] Step 2 Rewrite the system in standard form. [ left beginarrayl x+y+z=180 x+1.5 z= y+1.5 z=194 endarray right. ] Adding -1 times the second equation to the first equation produces a new first equation. [ left beginarrayl y-0.5 z= x+1.5 z= y+1.5 z=194 endarray right. ]
Transcript text: Find the values of $x, y$, and $z$ in the figure (in degrees), where $b=2$ and $c=14$. Step 1 Recall that in a triangle, the sum of the measures of an exterior angle and its adjacent angle is 180,180 . and the sum of the measures of the internal angles is $180^{\circ}$. Using the above facts and the given figure, write the system of linear equations. \[ \left\{\begin{array}{rlrl} x+y+z & =180 & & \text { Equation 1 } \\ x+1.5 z+2 & =180 \\ y+1.5 z-14 & =180 \end{array}, \quad \begin{array}{ll} 180 \end{array}\right. \] Step 2 Rewrite the system in standard form. \[ \left\{\begin{array}{l} x+y+z=180 \\ x+1.5 z=\square \\ y+1.5 z=194 \end{array}\right. \] Adding -1 times the second equation to the first equation produces a new first equation. \[ \left\{\begin{array}{l} y-0.5 z=\square \\ x+1.5 z=\square \\ y+1.5 z=194 \end{array}\right. \]
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Solution

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Solution Steps

Step 1: Write the system of linear equations

Using the given information and the properties of triangles, we can write the following system of linear equations:

  1. \( x + y + z = 180 \) (sum of internal angles in a triangle)
  2. \( x + 1.5z = 180 \) (sum of an exterior angle and its adjacent interior angle)
  3. \( y + 1.5z - 14 = 180 \) (sum of an exterior angle and its adjacent interior angle)
Step 2: Simplify the equations

Rewrite the equations in standard form:

  1. \( x + y + z = 180 \)
  2. \( x + 1.5z = 180 \)
  3. \( y + 1.5z = 194 \) (simplified from \( y + 1.5z - 14 = 180 \))
Step 3: Solve the system of equations

Subtract the second equation from the first: \[ (x + y + z) - (x + 1.5z) = 180 - 180 \] \[ y - 0.5z = 0 \] \[ y = 0.5z \]

Substitute \( y = 0.5z \) into the third equation: \[ 0.5z + 1.5z = 194 \] \[ 2z = 194 \] \[ z = 97 \]

Substitute \( z = 97 \) back into \( y = 0.5z \): \[ y = 0.5 \times 97 \] \[ y = 48.5 \]

Substitute \( y = 48.5 \) and \( z = 97 \) into the first equation: \[ x + 48.5 + 97 = 180 \] \[ x + 145.5 = 180 \] \[ x = 34.5 \]

Final Answer

The values of \( x \), \( y \), and \( z \) are: \[ x = 34.5^\circ \] \[ y = 48.5^\circ \] \[ z = 97^\circ \]

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