Problem Solving

If ABCDE is a regular pentagon, what is the value of x in the figure above?

A. 36
B. 42
C. 45
D. 48
E. 72

The OA is A.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.

AAPL wrote:

If ABCDE is a regular pentagon, what is the value of x in the figure above?

A. 36
B. 42
C. 45
D. 48
E. 72

The OA is A.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.

The total degrees of a polygon with n sides is (n-2)* 180.
Total degrees in a pentagon = (5-2)*180 = 3*180 = 540.
If all the angles are equal (as they are in a regular pentagon), then each angle is 540/5 = 108.

Now we know that the angles FEA and FAE will each be 180-108 = 72. If we have two angles measuring 72 degrees, which will sum to 144 degrees, we know the remaining angle will be 180-144 = 36. The answer is A

If ABCDE is a regular pentagon, what is the value of x in the figure above?

A. 36
B. 42
C. 45
D. 48
E. 72

The OA is A.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.

Hi AAPL,
Let's take a look at your question.

ABCDE is a regular pentagon. We know that we can calculate the sum of interior angles of a regular polygon using the formula:
$$\left(n-2\right)\times180^o$$
Where n is the number of sides of the polygon.

For a regular pentagon n = 5.
Therefore, sum of interior angles of a pentagon will be:
$$=\left(5-2\right)\times180^o$$
$$=\left(3\right)\times180^o=540^o$$

Since there are five angles inside the pentagon, we can calculate the measure of each angle as:
$$=\frac{540^o}{5}=108^o$$

Now we can find the interior angles of the triangle FAE.
$$m\angle FEA=180^o-m\angle DEA$$
$$m\angle FEA=180^o-108^o=72^o$$

Similarly,
$$m\angle FAE=180^o-m\angle BAE$$
$$m\angle FAE=180^o-108^o=72^o$$

Now we can find the value of x.
We know that sum of interior angles of a triangle is 180 degree, therefore,

$$x^o=180^o-m\angle FEA-m\angle FAE$$
$$x^o=180^o-72^o-72^o$$
$$x^o=36^o$$

Therefore, Option A is correct.

Hope it helps.
I am available if you'd like any follow up.