To solve for the measure of angle 2, we need to use the information provided about the measures of angle 1 and angle 3.
Given:
- The measure of angle 1 is [tex]\((10x + 8)^\circ\)[/tex].
- The measure of angle 3 is [tex]\((12x - 10)^\circ\)[/tex].
Assume that angles 1 and 3 are supplementary. Two angles are supplementary if their sum is [tex]\(180^\circ\)[/tex]. This assumption is reasonable based on standard geometric principles unless otherwise stated.
So, we set up the equation:
[tex]\[
(10x + 8) + (12x - 10) = 180
\][/tex]
1. Combine like terms:
[tex]\[
10x + 12x + 8 - 10 = 180
\][/tex]
2. Simplify the equation:
[tex]\[
22x - 2 = 180
\][/tex]
3. Add 2 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
22x = 182
\][/tex]
4. Divide both sides by 22:
[tex]\[
x = \frac{182}{22}
\][/tex]
[tex]\[
x = 8.2727\ldots \approx 8.273
\][/tex]
Now that we have the value of [tex]\(x\)[/tex], we can find the measures of angle 1 and angle 3.
5. Calculate the measure of angle 1:
[tex]\[
10x + 8 = 10(8.273) + 8 = 82.73 + 8 = 90.73^\circ
\][/tex]
6. Calculate the measure of angle 3:
[tex]\[
12x - 10 = 12(8.273) - 10 = 99.276 - 10 = 89.276^\circ \approx 89.28^\circ
\][/tex]
Since angles 1 and 2 are supplementary:
[tex]\[ \text{Measure of Angle 2} = 180^\circ - \text{Measure of Angle 1} \][/tex]
7. Calculate the measure of angle 2:
[tex]\[
\text{Measure of Angle 2} = 180^\circ - 90.73^\circ = 89.27^\circ
\][/tex]
Rounding to the nearest whole number, the measure of angle 2 is [tex]\(89^\circ\)[/tex]. Note: if we precisely measure the weighing system might give 98. This depends on exact measure assumptions. Be cautious of approximation which should ideally compensate our further calculations model.
