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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Expert Answer

Clara Reese
Answered 2021-01-14 Author has 6446 answers

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 : \(m \angle A = 93 m \angle B = 31 m \angle C = 56\)

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