Question

find the longest side of \\( \triangle abc \\), with \\( m \angle a = 60 ^ { \circ }, m \angle b = ( 2 x - 15 ) ^ { \circ } \\), and \\( m \angle c = ( 3 x + 10 ) ^ { \circ } \\).
the longest side of \\( \triangle abc \\) is \\( \overline { a b } \\). \\( \overline { b c } \\). \\( \overline { a c } \\).

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A+m\angle B + m\angle C=180^{\circ}\).
Substitute \(m\angle A = 60^{\circ}\), \(m\angle B=(2x - 15)^{\circ}\), and \(m\angle C=(3x + 10)^{\circ}\) into the equation:
\(60+(2x-15)+(3x + 10)=180\).
Simplify the left - hand side: \(60+2x-15 + 3x+10=180\), which becomes \(5x + 55=180\).
Subtract 55 from both sides: \(5x=180 - 55\), so \(5x=125\).
Divide both sides by 5: \(x = 25\).

Step2: Find the measures of \(\angle B\) and \(\angle C\)

For \(\angle B\): Substitute \(x = 25\) into \(m\angle B=(2x - 15)^{\circ}\).
\(m\angle B=(2\times25-15)^{\circ}=(50 - 15)^{\circ}=35^{\circ}\).
For \(\angle C\): Substitute \(x = 25\) into \(m\angle C=(3x + 10)^{\circ}\).
\(m\angle C=(3\times25+10)^{\circ}=(75 + 10)^{\circ}=85^{\circ}\).

Step3: Use the side - angle relationship in a triangle

In a triangle, the larger the angle, the longer the side opposite it.
Since \(m\angle C=85^{\circ}\) (the largest angle), the side opposite \(\angle C\) is \(\overline{AB}\).

Answer:

\(\overline{AB}\)