4. In the diagram, nine identical isosceles trapezoids are connected as shown to form a closed ring. (An isosceles trapezoid is a trapezoid with the properties that one pair of sides is parallel and the other two sides are equal in length.) What is the measure of $\angle A B C$ ?
4. Solution 1
The inner polygon formed by the nine trapezoids is a regular polygon with 9 sides. This is because (i) all of the sides of this polygon are equal in length (equal to the shorter parallel side of one of the trapezoids), and (ii) the measure of each of the interior angles of this polygon is equal (formed between two identical trapezoids).
The sum of the measures of the angles in this polygon with 9 sides is $180^{\circ}(9-2)$ or $1260^{\circ}$. Thus, the measure of each of the interior angles is $1260^{\circ} \div 9=140^{\circ}$.
Let $\theta=\angle A B C$.
Since the nine trapezoids are identical, then the trapezoid adjacent to the left also has an angle of $\theta$, as shown:

Since the sum of the measures of angles around a point is $360^{\circ}$, then $2 \theta+140^{\circ}=360^{\circ}$ and so $2 \theta=220^{\circ}$ which gives $\angle A B C=\theta=110^{\circ}$.

Solution 2
The outer polygon formed by the nine trapezoids is a regular polygon with 9 sides. This is because (i) all of the sides of this polygon are equal in length (equal to the longer parallel side of one of the trapezoids), and (ii) the measure of each of the interior angles of this polygon is equal (formed by two of the base angles of two identical trapezoids.)
The sum of the measures of the angles in this polygon with 9 sides is $180^{\circ}(9-2)$ or $1260^{\circ}$.
Thus, the measure of each of the interior angles is $1260^{\circ} \div 9=140^{\circ}$.
Each of these interior angles is made up from two equal base angles, which are thus equal to one-half of $140^{\circ}$ or $70^{\circ}$.
Since the trapezoids have parallel sides, then $\angle A B C$ is supplementary to one of these base angles, and so $\angle A B C=180^{\circ}-70^{\circ}=110^{\circ}$.
ANSWER: $110^{\circ}$