Vertical Angles and Straight Line Angles Calculation
The image depicts two angles formed by a straight line and a vertically opposite angle, which are always equal. The two angles given are:
1. \((4x + 2)^\circ\)
2. \((7x - 6)^\circ\)
Since they are vertically opposite angles, they are equal. Therefore, we can set up the following equation:
\(4x + 2 = 7x - 6\)
Now, let's solve for \(x\):
\(4x + 2 = 7x - 6\)
\(6 + 2 = 7x - 4x\)
\(8 = 3x\)
\(x = \frac{8}{3} \approx 2.67\)
Now we can use the value of \(x\) to find \(y\). Looking at the lines, we can see that they form a straight line and the sum of angles on a straight line is \(180^\circ\). This information can be used to form an equation using either of the two expressions for the angles given:
Using the angle \((4x + 2)^\circ\), we get:
\(4x + 2 + y = 180\)
Now substituting the value of \(x\) we found:
\(4(\frac{8}{3}) + 2 + y = 180\)
\( \frac{32}{3} + 2 + y = 180\)
\( \frac{38}{3} + y = 180\)
Multiply both sides by 3 to rid the fraction:
\(38 + 3y = 540\)
Now subtract 38 from both sides:
\(3y = 540 - 38\)
\(3y = 502\)
Divide both sides by 3 to solve for \(y\):
\(y = \frac{502}{3}\)
\(y = 167.\overline{333}\) or \(167\frac{1}{3}^\circ\)
Hence, the values are:
\(x \approx 2.67\)
\(y \approx 167.33^\circ\) or \(167\frac{1}{3}^\circ\)
