 # Challenge In triangleABC,mangleBtriangleABC,mangleB is one-thirdthe mangleAmangleA and mangleCmangleC is 37 less than themangleA.mangleA. What are the measures of theangles of triangleABC? Bevan Mcdonald 2021-01-13 Answered

Challenge In $$\triangle ABC,m\angle B$$ is one-third the $$m\angle A\ \text{and}\ m\angle C$$ is 37 less than the $$m\angle A.$$. What are the measures of the angles of $$\triangle ABC$$?

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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 $$( \frac{3}{7} ) m \angle A = 217 ( \frac{3}{7} ) = 31 \times 3 = 93 m \angle A = 93$$

compensate in ( 1) by $$m \angle A = 93 m < B = ( 1 / 3 ) \times 93 = ( 93 / 3 ) = 31$$ 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 : $$m \angle A = 93 m \angle B = 31 m \angle C = 56$$