Angles of triangle and the direct proportionality

Angles A, B and C of triangle ABC are directly proportional to numbers 3, 7 and 8, respectively.  Calculate the size of the smallest angle of triangle.

Solution:

We know that the directly proportional sizes form a string of equal ratios.

So:

(size of angle A)/3 = (size of angle B)/7 = (size of angle C)/8 = k (the constant of proportionality); 

                                                                                                                                      (relationship 1)

Note: (angle A)/3 = (angle A)-out-of-3 

Because the angles A, B and C are the angles of the triangle, these three angles always add to 180⁰.

(size of angle A) + (size of angle B) + (size of angle C) = 180 ⁰ ;   (the relationship 2)

Also, we know: from a string of equal ratios we can obtain a new ratio with the sum of the numerators-out-of-the sum of the denominators. Using the above relationship in 1 and 2 we obtain

(angle A)/3 = (angle B)/7 = (angle C)/8 = (angle + angle B + angle C)/(3 + 7 + 8) = 180 ⁰ /18 = 10 ⁰ = k (the constant of proportionality).

Now, we calculate the measures of the angles by matching each report with k.

(angle A)/3 = 10 ⁰; the measure of the angle A = 3*10 = 30 ⁰

(angle B)/7 = 10 ⁰; the measure of the angle B = 7*10 ⁰ = 70 ⁰

(angle C)/8 = 10 ⁰; the measure of angle C = 8*10 = 80 ⁰

The smallest angle in the triangle is ∡A =30 ⁰