Question - Vertical Angles and Straight Line Angles Calculation
Solution:
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 $$167\frac{1}{3}^\circ$$
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