Bevan Mcdonald

2021-01-13

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

Clara Reese

in triangle ABC

$m\mathrm{\angle }B=\left(\frac{1}{3}\right)m\mathrm{\angle }A$ --------- ( 1 )

$m\mathrm{\angle }C=m\mathrm{\angle }A-37$ ------------ ( 2 )

the sum of the internal angles of a triangle = 180 thus

$m\mathrm{\angle }A+m\mathrm{\angle }B+m\mathrm{\angle }C=180$ ----------- ( 3 )

compensate from ( 1 ) and ( 2 ) in ( 3 ) $m\mathrm{\angle }A+\left(\frac{1}{3}\right)m\mathrm{\angle }A+\left(m\mathrm{\angle }A-37\right)=180\left(\frac{7}{3}\right)m\mathrm{\angle }A-37=180$ ( add 37 to both sides ) $\left(\frac{7}{3}\right)m\mathrm{\angle }A=180+37=217$ ( multiply both sides by $\left(\frac{3}{7}\right)m\mathrm{\angle }A=217\left(\frac{3}{7}\right)=31×3=93m\mathrm{\angle }A=93$

compensate in ( 1) by $m\mathrm{\angle }A=93m compensate in ( 2 ) by $m thus $m\mathrm{\angle }A=93m\mathrm{\angle }B=31m

answer measures of the angles of the triangle ABC : $m\mathrm{\angle }A=93m\mathrm{\angle }B=31m\mathrm{\angle }C=56$

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