5. In an isosceles triangle $A B C$ the altitude to the leg $\overline{B C}$ and the altitude to the base $\overline{A B}$ intersect at point $E$ such that $|C E|=|A B|$. Determine the measures of the interior angles of triangle $A B C$.

NATIONAL MATHEMATICS COMPETITIONŠibenik, April 2-4, 2014.
5. 

![](https://cdn.mathpix.com/cropped/2024_05_30_3b96b040deecb25706d7g-06.jpg?height=620&width=551&top_left_y=1523&top_left_x=364)

It holds that $|\angle ECD| = |\angle BAD|$ because these are acute angles with mutually perpendicular legs.

Furthermore, $|\angle CDE| = 90^\circ = |\angle ADB|$ and $|CE| = |AB|$ according to the conditions of the problem.

Therefore, $|\angle DEC| = |\angle DBA|$ and by the Side-Angle-Side (SAS) congruence theorem, $\triangle CED \cong \triangle ABD$. From the congruence, it follows that $|CD| = |AD|$, so $\triangle ADC$ is isosceles and right-angled.

The measures of the acute angles in $\triangle ADC$ are $45^\circ$, so $|\angle ACB| = 45^\circ$.

Finally, $|\angle BAC| = |\angle CBA| = \frac{180^\circ - 45^\circ}{2} = 67.5^\circ$.

## NATIONAL MATHEMATICS COMPETITION

Šibenik, April 2-4, 2014.