Question

1. two congruent regular pentagons share common side as shown below.

image of two regular pentagons sharing a common side ab
what is the measure of angle, likely ∠abc or similar?
a. 72°
b. 108°
c. 36°
d. 144°

Explanation:

Step1: Find interior angle of regular pentagon

The formula for the interior angle of a regular polygon is $\frac{(n - 2)\times180^{\circ}}{n}$, where $n = 5$ for a pentagon.
$\frac{(5 - 2)\times180^{\circ}}{5}=\frac{3\times180^{\circ}}{5}=108^{\circ}$.

Step2: Calculate $\angle ABC$

The sum of angles around point $B$ (or $A$) on a straight line? Wait, actually, around point $B$, the two pentagons have angles at $B$ (from each pentagon) and $\angle ABC$. The sum of angles around a point is $360^{\circ}$, but here, since the two pentagons are congruent and share a side, the angles at $B$ from each pentagon are $108^{\circ}$ each. So $\angle ABC=360^{\circ}- 108^{\circ}-108^{\circ}=144^{\circ}$? Wait, no, wait. Wait, maybe the angle in question is the angle between the two pentagons. Wait, maybe I misread. Wait, the problem is about two congruent regular pentagons sharing a common side, so the angle between the non - shared sides. Wait, let's re - evaluate.

Wait, the interior angle of a regular pentagon is $108^{\circ}$. When two regular pentagons share a common side, the angle between the two adjacent sides (from each pentagon) at the common vertex (like $B$) can be found by considering the linear pair or the sum around the point. Wait, the sum of angles around a point is $360^{\circ}$. If we have two interior angles of the pentagons (each $108^{\circ}$) at point $B$, then the angle $\angle ABC$ (the angle we need to find) is $360^{\circ}-108^{\circ}-108^{\circ}=144^{\circ}$? But wait, the options have $144^{\circ}$ as option D. Wait, but let's check again.

Wait, maybe the angle is the angle between the two sides that are not the common side. Let's confirm the interior angle of a regular pentagon: $\frac{(5 - 2)\times180}{5}=108^{\circ}$. Then, at the vertex where the two pentagons meet (e.g., point $B$), the two angles from the pentagons are each $108^{\circ}$. So the remaining angle (the angle we need) is $360 - 108 - 108=144^{\circ}$.

Answer:

D. $144^{\circ}$