Question

У трикутнику АВС кут А=80 , кут С=40 . Під яким кутом перетинаються висоти АМ і СN трикутника АВС ?

Answer 1

Трикутник АВС, СН і АМ - висоти, кут А=80, кутС=40, трикутник АМС прямокутний, кут МАС= 90-кутС=90-40=50, кут ВАМ=кутА-кут МАС=80-50=30, трикутник АОН прямокутний, точка О перетин висот, кут АОН=90-кут ВАМ=90-30=60=кутМОС, кут НОМ=кутАОС=180-60=120

Answer 2

Finding the Angle of Intersection of the Altitudes

To find the angle at which the altitudes AM and CN intersect in triangle ABC, we can use the properties of triangles and angles.

Given that angle A is 80 degrees and angle C is 40 degrees, we can determine the measure of angle B using the fact that the sum of the angles in a triangle is 180 degrees.

To find angle B: 1. Subtract the measures of angles A and C from 180 degrees: 180 - 80 - 40 = 60 degrees.

Now that we know the measure of angle B, we can proceed to find the angle at which the altitudes AM and CN intersect.

Let's denote the point of intersection of the altitudes as point O.

To find the measure of angle MON (where M is the foot of the altitude from A and N is the foot of the altitude from C), we can use the fact that the intersection of two lines creates vertical angles.

Vertical angles are congruent, meaning they have the same measure. Therefore, angle MON is equal to angle B.

Therefore, the angle at which the altitudes AM and CN intersect in triangle ABC is 60 degrees.

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