25. In the isosceles triangle $A B C$, points $K$ and $L$ are marked on the equal sides $A B$ and $B C$ respectively so that $A K=K L=L B$ and $K B=A C$. What is the size of angle $A B C$ ?
A $36^{\circ}$
B $38^{\circ}$
C $40^{\circ}$
D $42^{\circ}$
E $44^{\circ}$
25. A Since triangle $A B C$ is isosceles with $A B=B C$ and we are given that $L B=A K$, the other parts of the equal sides must themselves be equal. Hence $L C=B K=A C$. Draw in line $K C$ as shown to form triangles $A C K$ and $L C K$. Since $A K=K L, A C=L C$ and $K C$ is common to both, triangles $A C K$ and $L C K$ are congruent and hence $\angle K A C=\angle C L K$.

Let the size of $\angle L B K$ be $x^{\circ}$. Since $K L=L B$, triangle $K L B$ is isosceles and hence $\angle B K L=x^{\circ}$. Since an exterior angle of a triangle is equal to the sum of the interior opposite angles, $\angle K L C=2 x^{\circ}$ and hence $\angle K A C=2 x^{\circ}$. Since the base angles of an isosceles triangle are equal, $\angle A C L=2 x^{\circ}$. Therefore, since angles in a triangle add to $180^{\circ}$, when we consider triangle $A B C$, we have $x^{\circ}+2 x^{\circ}+2 x^{\circ}=180^{\circ}$ and hence $x=36$. Therefore the size of $\angle A B C$ is $36^{\circ}$.