# Challenge In triangleABC,mangleBtriangleABC,mangleB is one-third the mangleAmangleA and mangleCmangleC is 37 less than the mangleA.mangleA. What are the measures of the angles of triangleABC?

Question
Challenge In \triangleABC,m\angleB\triangleABC,m\angleB is one-third the m\angleAm\angleA and m\angleCm\angleC is 37 less than the m\angleA.m\angleA. What are the measures of the angles of \triangleABC?

2021-01-14
in triangle ABC m \angle B = ( \frac{1}{3} ) m \angle A --------- ( 1 ) m \angle C = m \angle A - 37 ------------ ( 2 ) the sum of the internal angles of a triangle = 180 thus m \angle A + m \angle B + m \angle C = 180 ----------- ( 3 ) compensate from ( 1 ) and ( 2 ) in ( 3 ) m \angle A + ( \frac{1}{3} ) m \angle A + ( m \angle A - 37 ) = 180 ( \frac{7}{3} ) m \angle A - 37 = 180 ( add 37 to both sides ) ( \frac{7}{3} ) m \angle A = 180 + 37 = 217 ( multiply both sides by ( \frac{3}{7} ) m \angle A = 217 ( \frac{3}{7} ) = 31 x 3 = 93 m \angle A = 93 compensate in ( 1) by m \angle A = 93 m < B = ( 1 / 3 ) x 93 = ( 93 / 3 ) = 31 compensate in ( 2 ) by m < A = 93 m \angle C = 93 - 37 = 56 thus m \angle A = 93 m \angle B = 31 m < C = 56 answer measures of the angles of the triangle ABC : m \angle A = 93 m \angle B = 31 m \angle C = 56

### Relevant Questions

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
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