Chapter 7.3, Problem 61E

To determine

To find: The values of $x,y,\text{and }z$ in the given figure.

Answer to Problem 61E

The values of $x,y,\text{and }z$ in the given figure are $60°,67°,\text{and }53°$ , respectively.

Explanation of Solution

Given information:

The triangle figure is given below:

  

Concept used:

Angle Sum property of a triangle says that the sum of interior angles of the triangle is $180°.$

Exterior angle theorem says that the exterior angle measure at a vertex is equal to the sum of the measures of its opposite interior angles.

Calculations:

From the given figure, we obtain

  $\begin{array}{l}x+y+z=180(\because \text{Angles sum property of a triangle})\\ x+y=2x+7(\because \text{Exterior angle theorem})\\ x+z=2x-7(\because \text{Exterior angle theorem})\end{array}$

Simplifying,

  $\begin{array}{l}x+y+z=180\to (1)\\ -x+y=7\to (2)\\ -x+z=-7\to (3)\end{array}$

The steps of solving the system of equations using Row-Echelon Form and Back substitutions are as follows:

Adding the first equation to the second equation to produce a new second equation and adding the first equation to the third equation to produce a new third equation, we get

  $\{\begin{array}{c}x+y+z=180\\ 2y+z=187\\ y+2z=173\end{array}$

Now, adding $-2$ times the third equation to the second equation to produce a new third equation, we get

  $\{\begin{array}{c}x+y+z=180\\ 2y+z=187\\ -3z=-159\end{array}$

Solving $-3z=-159$ for $z$ , we obtain $z=53$ .

Substituting this value in the second equation, we get

  $\begin{array}{l}2y+53=187\\ \Rightarrow 2y=134\\ \Rightarrow y=67\end{array}$

Now, substituting the above finding values in the first equation, we get

  $\begin{array}{l}x+67+53=180\\ \Rightarrow x+120=180\\ \Rightarrow x=60\end{array}$

Conclusion:

Therefore, the values of $x,y,\text{and }z$ in the given figure are $60°,67°,\text{and }53°$ , respectively.