## Task B-1.4.

Points $F, G$, and $H$ lie on the side $\overline{A B}$ of triangle $A B C$. Point $F$ is between points $A$ and $G$, and point $H$ is between points $G$ and $B$. The measure of angle $C A B$ is $5^{\circ}$, and $|B H|=|B C|,|H G|=$ $|H C|,|G F|=|G C|,|F A|=|F C|$. What is the measure of angle $A B C$?
## Solution.

Let's draw triangle $A B C$ and consider triangle $A F C$.

![](https://cdn.mathpix.com/cropped/2024_05_30_a25c851e6461b5bf7f1cg-03.jpg?height=205&width=1120&top_left_y=1945&top_left_x=388)

From the given condition $(|F A|=|F C|)$, triangle $A F C$ is isosceles. One angle is $5^{\circ}$, so the angle $\varangle A F C=170^{\circ}$.

Then $\varangle B F C=10^{\circ}$, and in the isosceles triangle $F G C$, $\varangle F G C=160^{\circ}$. 1 point

In the isosceles triangle $G H C$, $\varangle C G H=20^{\circ}$, i.e., $\varangle G H C=140^{\circ}$ 1 point and $\varangle C H B=40^{\circ}$. 1 point

Finally, triangle $C H B$ is isosceles, and the desired angle is $\varangle A B C=\varangle H B C=100^{\circ} . \quad 1$ point