\left.\begin{array}{l}\text { [Sum of angles in a triangle. Theorem about the exterior angle.] } \\ {[\text { Concurrency of altitudes. Angles between altitudes. ] }}\end{array}\right]

The altitudes of triangle $ABC$, drawn from vertices $A$ and $C$, intersect at point $M$. Find $\angle AMC$, if $\angle A=$ $70^{\circ}, \angle C=80^{\circ}$.
Find the angle between side $B C$ and the height drawn from vertex $C$.

## Solution

Let $A A_{1}$ and $C C_{1}$ be the heights.

$\angle A C C_{1}=90^{\circ}-70^{\circ}=20^{\circ}, \angle C A A_{1}=90^{\circ}-80^{\circ}=10^{\circ}, \angle A M C=180^{\circ}-20^{\circ}-10^{\circ}=150^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_837b6360ec703a44a031g-20.jpg?height=425&width=440&top_left_y=1474&top_left_x=818)

## Answer

$150^{\circ}$.