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I need help with this exercise.

"Using the figure below, determine the measure of the interior angle at vertex A."

enter image description here

Choose one:

a. $60^\circ$ b. $150^\circ$ c. $300^\circ$ d. $150^\circ$

I would like to know if what I did is right.

Since polygon has a 7 sides the sum of the interior angles is $(7-2)180^\circ=900^\circ$.

Then,

$$2x+2x+6x+5x+5x+5x+5x=900$$ $$30x=900$$ $$x=30$$

Then,

Angle with vertex $A$ has a measure $5x=5(30)=150^\circ$.

Is this ok?

Appreciate the help in advance.

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  • $\begingroup$ Yes this is correct. The interior angles of a polygon with 7 sides is always 900°. And the calculation is correct as well. The answer is d). $\endgroup$
    – Saha
    Mar 2, 2022 at 15:21
  • $\begingroup$ Someone submitted an edit to change $60\degree$ to $60°.$ Rather than approve that edit as-is, I clicked on "improve edit" and changed it to $60^\circ,$ coded as 60^\circ. Are there other opinions as to which looks better (the two options being $60°$ and $60^\circ$)? $\endgroup$ Mar 2, 2022 at 15:25
  • $\begingroup$ Perhaps worthy of note is that the angle given as $6x$ has measure $180^\circ$ so is actually a straight line and the polygon has only 6 sides. Regardless, the answer is the same. Also, are answer options b. and d. both meant to be $150^\circ$? $\endgroup$
    – nickgard
    Mar 2, 2022 at 23:29

1 Answer 1

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$\newcommand{\d}{^\circ}$Imagine driving a car counterclockwise around this circuit. At angle $A,$ you turn $180\d-5x$ to the right.

Add up all your right turns as you go around, counting a turn to the left as a negative number of degrees to the right. To complete one full circuit, you turn $360\d$ to the right. So \begin{align} 360\d & = 180\d-5x \\ & {} + 180\d-5x \\ & {} + 180\d-5x \\ & {} + 180\d-2x \\ & {} + 180\d-6x \\ & {} + 180\d-2x \\ & {} + 180\d - 5x \end{align} Solve that for $x.$

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  • $\begingroup$ This is equivalent to solving $2x+2x+6x+5x+5x+5x+5x=900°$ and is thus the same answer OP already proposed. $\endgroup$
    – Saha
    Mar 2, 2022 at 17:18
  • $\begingroup$ @Saha : But it also explains a reason why that should be the sum: the sum of the turns should be $360^\circ. \qquad$ $\endgroup$ Mar 2, 2022 at 18:03
  • $\begingroup$ Ok, fair enough. $\endgroup$
    – Saha
    Mar 2, 2022 at 18:06

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