To find the missing angle measurements in the table, let's go step by step through the problem with the given information.
We are given the following incomplete table and need to solve for missing values:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
40 & 140 & 40 & 95 \\
\hline
& \overline{x-4} & 35 & 11 \\
\hline
& & x+2 + x+2 & 97 \\
\hline
\end{tabular}
\][/tex]
### Step 1: Determine [tex]\(x\)[/tex]
We start by recognizing that the sum of angles in each row should be 180 degrees. Let's call the missing value in row 2, column 2, [tex]\(x\)[/tex]. Consequently, the angle in row 2, column 2 should be [tex]\(\overline{x-4}\)[/tex].
Given the column has a negative angle, let’s think about possible values for x
To find [tex]\(x\)[/tex], we write the equation for the sum of the second row's angles:
[tex]\[ \overline{x-4} + 35 + 11 + 4 = 180 \][/tex]
[tex]\[ \overline{x-4} + 50 = 180 \][/tex]
Solving for [tex]\(\overline{x-4}\)[/tex]:
[tex]\[ \overline{x-4} = 180 - 50 \][/tex]
[tex]\[ \overline{x-4} = 130 \][/tex]
So, to maintain consistency, [tex]\(x = -130\)[/tex].
### Step 2: Determine Values in the Third Row
We're provided that the angle in row 3, column 3 is [tex]\(x+2 + x+2\)[/tex] and the angle in row 3, column 4 is 97.
Let's calculate:
[tex]\[ x+2 + x+2 = -130+2 + (-130) + 2 = -256\][/tex]
Thus the angle in row 3, column 3 is -256.
To find row 3, column 1, again, we use the sum of row 3's angles which should equal 180 degrees:
[tex]\[ \overline{\text{row 3, col 1}} + \overline{\text{row 3, col 3}} + 97 = 180 \][/tex]
So:
[tex]\[ row 3, column 1= 180 - (-256) - 97 \][/tex]
[tex]\[ row 3, column 1= 180 - (-353)\][/tex]
[tex]\[ row 3 column 1= 339\][/tex]
### Solution Summary:
- [tex]\(x = -130\)[/tex]
- [tex]\(\text{row 3, column 3} =-256\)[/tex]
- [tex]\(\text{row 3, column 1} =339 \)[/tex]
So, the missing angle measurements are:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
40 & 140 & 40 & 95 \\
\hline
& -130 & 35 & 11 \\
\hline
339 & &-256& 97 \\
\hline
\end{tabular}
\][/tex]
