Problem 2. In an acute-angled triangle, three of the six adjacent angles formed around the orthocenter by the lines including the three altitudes have measures proportional to the numbers 5, 5, and 7, and the sum of the measures of the other three angles is equal to $190^{\circ}$. Determine the measures of the angles of the triangle.
Problem 2. In an acute-angled triangle, three of the six angles formed around the orthocenter by the lines that include the three altitudes have measures proportional to the numbers 5, 5, and 7, and the sum of the measures of the other three angles is equal to $190^{\circ}$. Determine the measures of the angles of the triangle.

## Solution

The sum of the measures of the angles proportional to 5, 5, and 7 is $170^{\circ}$. .........1p The measures of these angles are $5x, 5x, 7x$, with $5x + 5x + 7x = 170^{\circ}$, from which $x = 10^{\circ}$. Around the orthocenter, there are three pairs of vertically opposite angles, hence congruent. On one side of each line containing an altitude, there is one angle from each pair. We know that three of the angles have measures $50^{\circ}, 50^{\circ}$, $70^{\circ}$, so the other three have measures $70^{\circ}, 60^{\circ}, 60^{\circ}$.

$3 p$

Using the measures of these angles, the measures of the angles formed by the altitudes with the sides are determined to be $20^{\circ}, 30^{\circ}, 40^{\circ}$, so the measures of the angles of the triangle are $50^{\circ}, 60^{\circ}, 70^{\circ}$. $.3 p$