# Sum of interior angles of a polygon

I need help with this exercise.

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

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?

• 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).
– Saha
Mar 2, 2022 at 15:21
• 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 $60^\circ$)? Mar 2, 2022 at 15:25
• 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 $150^\circ$? Mar 2, 2022 at 23:29

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